Class 12th Mathematics Part I CBSE Solution
Exercise 3.1- In a matrix A = a^n = [cccc 2&5&19&-7 35&-2& 5/2 &12 root 3 &1&-5&17] , (i) The…
- If a matrix has 24 elements, what are the possible orders it can have? What, if…
- If a matrix has 18 elements, what are the possible orders it can have? What, if…
- a_ij = (i+j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
- a_ij = i/j Construct a 2 × 2 matrix, A = [aij], whose elements are given by:…
- a_ij = (i+2j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
- a_ij = 1/2 |-3i+j| Construct a 3 × 4 matrix, whose elements are given by:…
- a_ij = 2i-j Construct a 3 × 4 matrix, whose elements are given by:…
- [ll 4&3 x&5] = [ll y 1&5] Find the values of x, y and z from the following…
- [cc x+y&2 5+z] = [ll 6&2 5&8] Find the values of x, y and z from the following…
- [c x+y+z x+z y+z] = [9 5 7] Find the values of x, y and z from the following…
- Find the value of a, b, c and d from the equation: [cc a-b&2a+c 2a-b&3c+d] = [cc…
- A = [aij]m × n is a square matrix, ifA. m n B. m n C. m = n D. None of these…
- Which of the given values of x and y make the following pair of matrices equal…
- The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:A.…
Exercise 3.2- Le t a = [ll 2&4 3&2] , b = [cc 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- [cc a -b] + [ll a b] Compute the following:
- [cc a^2 + b^2 & b^2 + c^2 a^2 + c^2 & a^2 + b^2] + [cc 2ab&2bc -2ac&-2ab]…
- [ccc -1&4&-6 8&5&16 2&8&5] + [ccc 12&7&6 8&0&5 3&2&4] Compute the following:…
- [cc cos^2x^2x sin^2x^2x] + [cc sin^2x^2x cos^2x^2x] Compute the following:…
- Compute the indicated products.
- [1 2 3] [lll 2&3&4] Compute the indicated products.
- [cc 1&-2 2&3] [ccc 1&2&3 2&3&1] Compute the indicated products.
- [ccc 2&3&4 3&4&5 4&5&6] [ccc 1&-3&5 0&2&4 3&0&5] Compute the indicated…
- Compute the indicated products.
- [ccc 3&-1&3 -1&0&2] [cc 2&-3 1&0 3&1] Compute the indicated products.…
- If a = [ccc 1&2&-3 5&0&2 1&-1&1] , b = [ccc 3&-1&2 4&2&5 2&0&3] c = [ccc 4&1&2…
- If a = [ccc 2/3 &1& 5/3 1/3 & 2/3 & 4/3 7/3 &2& 2/3] b = [ccc 2/5 & 3/5 &1 1/5 &…
- Simplify costheta [cc costheta heta -sintegrate heta]+sintegrate heta [cc…
- x+y = [ll 7&0 2&5] x-y = [ll 3&0 0&3] Find X and Y, if
- 2x+3y = [ll 2&3 4&0] 3x+2y = [cc 2&-2 -1&5] Find X and Y, if
- Find X, if y = [ll 3&2 1&4] 2x+y = [cc 1&0 -3&2]
- Find x and y, if 2 [ll 1&3 0] + [ll y&0 1&2] = [ll 5&6 1&8]
- Solve the equation for x, y, z and t, if 2 [ll x y]+3 [cc 1&-1 0&2] = 3 [cc 3&5…
- If x [2 3]+y [c -1 1] = [c 10 5] , find the values of x and y.
- Given 3 [cc x z] = [cc x&6 -1&2w] + [cc 4+y z+w&3] , find the values of x, y, z…
- If f (x) = [ll cosx&-sinx&0 sinx&0 0&0&1 show that F(x) F(y) = F(x + y).…
- [cc 5&-1 6&7] [cc 2&1 3&4] not equal [ll 2&1 3&4] [cc 5&-1 6&7] Show that…
- [ccc 1&2&3 0&1&0 1&1&0] [ccc -1&1&0 0&-1&1 2&3&4] = [ccc -1&1&0 0&-1&1 2&3&4]…
- Find A^2 - 5A + 6I, if a = [lll 2&0&1 2&1&3 1&-1&0]
- If a = [lll 1&0&2 0&2&1 2&0&3] , prove that A3 - 6A^2 + 7A + 2I = 0…
- If a = [ll 3&-2 4&-2] i = [ll 1&0 0&1] , find k so that A^2 = kA - 2I…
- If a = [ccc 0& - tan alpha /2 tan alpha /2 &0] and I is the identity matrix of…
- Rs 1800 A trust fund has Rs. 30,000 that must be invested in two different…
- Rs. 2000 A trust fund has Rs. 30,000 that must be invested in two different…
- The bookshop of a particular school has 10 dozen chemistry books, 8 dozen…
- The restriction on n, k and p so that PY + WY will be defined are:A. k = 3, p =…
- If n = p, then the order of the matrix 7X - 5Z is:A. p × 2 B. 2 × n C. n × 3 D.…
Exercise 3.3- [c 5 1/2 -1] Find the transpose of each of the following matrices:…
- [cc 1&-1 2&3] Find the transpose of each of the following matrices:…
- [ccc -1&5&6 root 3 &5&6 2&3&-1] Find the transpose of each of the following…
- If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A + B)’ = A’ + B’,…
- If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A - B)’ = A’- B’…
- If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A + B). = A’…
- If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A - B)’ =…
- If a = [rr -2&3 1&2] and b = [rr -1&0 1&2] , then find (A + 2B)’
- (i) a = [c 1 -4 3] b = [lll -1&2&1] For the matrices A and B, verify that (AB)’…
- b = [lll 1&5&7]a = [0 1 2] For the matrices A and B, verify that (AB)’ = B’A’,…
- If a = [ll cosalpha -sinalpha] , then verify that A’ A = I
- If a = [cc sinalpha -cosalpha] , then verify that A’ A = I
- Show that the matrix a = [ccc 1&-1&5 -1&2&1 5&1&3] is a symmetric matrix.…
- Show that the matrix a = [ccc 0&1&-1 -1&0&1 1&-1&0] is a skew symmetric matrix.…
- For the matrix a = [ll 1&5 6&7] , verify that (A + A’) is a symmetric matrix…
- For the matrix a = [ll 1&5 6&7] , verify that (A - A’) is a skew symmetric…
- Find and 1/2 (a-a^there there eξ sts) , when a = [ccc 0 -a&0 -b&-c&0]…
- [cc 3&5 1&-1] Express the following matrices as the sum of a symmetric and a…
- [ccc 6&-2&2 -2&3&-1 2&-1&3] Express the following matrices as the sum of a…
- [ccc 3&3&-1 -2&-2&1 -4&-5&2] Express the following matrices as the sum of a…
- [cc 1&5 -1&2] [cc 3&5 1&-1] Express the following matrices as the sum of a…
- If A, B are symmetric matrices of same order, then AB - BA is aA. Skew…
- If , and A + A’ = I, if the value of a isA. 3 pi /2 B. pi /3 C. pi D. 3 pi /2…
Exercise 3.4- Using elementary transformations, find the inverse of each of the matrices.…
- [ll 2&1 1&1] Using elementary transformations, find the inverse of each of the…
- [ll 1&3 2&7] Using elementary transformations, find the inverse of each of the…
- [ll 2&3 5&7] Using elementary transformations, find the inverse of each of the…
- [ll 2&1 7&4] Using elementary transformations, find the inverse of each of the…
- [ll 2&5 1&3] Using elementary transformations, find the inverse of each of the…
- [ll 3&1 5&2] Using elementary transformations, find the inverse of each of the…
- [ll 4&5 3&4] Using elementary transformations, find the inverse of each of the…
- [cc 3&10 2&7] Using elementary transformations, find the inverse of each of the…
- [cc 3&-1 -4&2] Using elementary transformations, find the inverse of each of…
- [ll 2&-6 1&-2] Using elementary transformations, find the inverse of each of…
- [cc 6&-3 -2&1] Using elementary transformations, find the inverse of each of…
- [cc 2&-1 -3&2] Using elementary transformations, find the inverse of each of…
- [ll 2&1 4&2] Using elementary transformations, find the inverse of each of the…
- [ccc 2&-3&3 2&2&3 3&-2&2] Using elementary transformations, find the inverse of…
- [ccc 1&3&-2 -3&0&-5 2&5&0] Using elementary transformations, find the inverse…
- [ccc 2&0&-1 5&1&0 0&1&3] Using elementary transformations, find the inverse of…
- Matrices A and B will be inverse of each other only ifA. AB = BA B. AB = BA = 0…
Miscellaneous Exercise- Let, show that (aI + bA)n = anI + nan-1bA, where I is the identity matrix of…
- If a = [lll 1&1&1 1&1&1 1&1&1] , prove that .
- If a = [ll 3&-4 1&-1] then prove that a^n = [cc 1+2n&-4n n&1-2n] where n is any…
- If A and B are symmetric matrices, prove that AB - BA is a skew symmetric…
- Show that the matrix B’AB is symmetric or skew symmetric according as A is…
- Find the values of x, y, z if the matrix satisfy a = [ccc 0&2y x&-z x&-y] the…
- For what values of x: [lll 1&2&1] [lll 1&2&0 2&0&1 1&0&2] [0 2 x] = 0?…
- If a = [cc 3&1 -1&2] show that A^2 - 5A + 7I =0.
- Find x, if [rrr x&-5&-1] [lll 1&0&2 0&2&1 2&0&3] [x 4 1] = 0
- A manufacturer produces three products x, y, z which he sells in two markets.…
- Find the matrix X so that x [ccc 1&2&3 4&5&6] = [ccc -7&-8&-9 2&4&6]…
- If A and B are square matrices of the same order such that AB = BA, then prove…
- If a = [ll alpha & beta gamma & - alpha] is such that A � = I, thenA. 1 +…
- If the matrix A is both symmetric and skew symmetric, thenA. A is a diagonal…
- If A is square matrix such that A^2 = A, then (I + A)^3 7 A is equal toA. A B.…
- In a matrix A = a^n = [cccc 2&5&19&-7 35&-2& 5/2 &12 root 3 &1&-5&17] , (i) The…
- If a matrix has 24 elements, what are the possible orders it can have? What, if…
- If a matrix has 18 elements, what are the possible orders it can have? What, if…
- a_ij = (i+j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
- a_ij = i/j Construct a 2 × 2 matrix, A = [aij], whose elements are given by:…
- a_ij = (i+2j)^2/2 Construct a 2 × 2 matrix, A = [aij], whose elements are given…
- a_ij = 1/2 |-3i+j| Construct a 3 × 4 matrix, whose elements are given by:…
- a_ij = 2i-j Construct a 3 × 4 matrix, whose elements are given by:…
- [ll 4&3 x&5] = [ll y 1&5] Find the values of x, y and z from the following…
- [cc x+y&2 5+z] = [ll 6&2 5&8] Find the values of x, y and z from the following…
- [c x+y+z x+z y+z] = [9 5 7] Find the values of x, y and z from the following…
- Find the value of a, b, c and d from the equation: [cc a-b&2a+c 2a-b&3c+d] = [cc…
- A = [aij]m × n is a square matrix, ifA. m n B. m n C. m = n D. None of these…
- Which of the given values of x and y make the following pair of matrices equal…
- The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:A.…
- Le t a = [ll 2&4 3&2] , b = [cc 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- Let a = [ll 2&4 3&2] , b = [rr 1&3 -2&5] , c = [cc -2&5 3&4] Find each of the…
- [cc a -b] + [ll a b] Compute the following:
- [cc a^2 + b^2 & b^2 + c^2 a^2 + c^2 & a^2 + b^2] + [cc 2ab&2bc -2ac&-2ab]…
- [ccc -1&4&-6 8&5&16 2&8&5] + [ccc 12&7&6 8&0&5 3&2&4] Compute the following:…
- [cc cos^2x^2x sin^2x^2x] + [cc sin^2x^2x cos^2x^2x] Compute the following:…
- Compute the indicated products.
- [1 2 3] [lll 2&3&4] Compute the indicated products.
- [cc 1&-2 2&3] [ccc 1&2&3 2&3&1] Compute the indicated products.
- [ccc 2&3&4 3&4&5 4&5&6] [ccc 1&-3&5 0&2&4 3&0&5] Compute the indicated…
- Compute the indicated products.
- [ccc 3&-1&3 -1&0&2] [cc 2&-3 1&0 3&1] Compute the indicated products.…
- If a = [ccc 1&2&-3 5&0&2 1&-1&1] , b = [ccc 3&-1&2 4&2&5 2&0&3] c = [ccc 4&1&2…
- If a = [ccc 2/3 &1& 5/3 1/3 & 2/3 & 4/3 7/3 &2& 2/3] b = [ccc 2/5 & 3/5 &1 1/5 &…
- Simplify costheta [cc costheta heta -sintegrate heta]+sintegrate heta [cc…
- x+y = [ll 7&0 2&5] x-y = [ll 3&0 0&3] Find X and Y, if
- 2x+3y = [ll 2&3 4&0] 3x+2y = [cc 2&-2 -1&5] Find X and Y, if
- Find X, if y = [ll 3&2 1&4] 2x+y = [cc 1&0 -3&2]
- Find x and y, if 2 [ll 1&3 0] + [ll y&0 1&2] = [ll 5&6 1&8]
- Solve the equation for x, y, z and t, if 2 [ll x y]+3 [cc 1&-1 0&2] = 3 [cc 3&5…
- If x [2 3]+y [c -1 1] = [c 10 5] , find the values of x and y.
- Given 3 [cc x z] = [cc x&6 -1&2w] + [cc 4+y z+w&3] , find the values of x, y, z…
- If f (x) = [ll cosx&-sinx&0 sinx&0 0&0&1 show that F(x) F(y) = F(x + y).…
- [cc 5&-1 6&7] [cc 2&1 3&4] not equal [ll 2&1 3&4] [cc 5&-1 6&7] Show that…
- [ccc 1&2&3 0&1&0 1&1&0] [ccc -1&1&0 0&-1&1 2&3&4] = [ccc -1&1&0 0&-1&1 2&3&4]…
- Find A^2 - 5A + 6I, if a = [lll 2&0&1 2&1&3 1&-1&0]
- If a = [lll 1&0&2 0&2&1 2&0&3] , prove that A3 - 6A^2 + 7A + 2I = 0…
- If a = [ll 3&-2 4&-2] i = [ll 1&0 0&1] , find k so that A^2 = kA - 2I…
- If a = [ccc 0& - tan alpha /2 tan alpha /2 &0] and I is the identity matrix of…
- Rs 1800 A trust fund has Rs. 30,000 that must be invested in two different…
- Rs. 2000 A trust fund has Rs. 30,000 that must be invested in two different…
- The bookshop of a particular school has 10 dozen chemistry books, 8 dozen…
- The restriction on n, k and p so that PY + WY will be defined are:A. k = 3, p =…
- If n = p, then the order of the matrix 7X - 5Z is:A. p × 2 B. 2 × n C. n × 3 D.…
- [c 5 1/2 -1] Find the transpose of each of the following matrices:…
- [cc 1&-1 2&3] Find the transpose of each of the following matrices:…
- [ccc -1&5&6 root 3 &5&6 2&3&-1] Find the transpose of each of the following…
- If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A + B)’ = A’ + B’,…
- If a = [ccc -1&2&3 5&7&9 -2&1&1] and, then verify that (A - B)’ = A’- B’…
- If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A + B). = A’…
- If a^there there eξ sts = [cc 3&4 -1&2 0&1] and, then verify that (A - B)’ =…
- If a = [rr -2&3 1&2] and b = [rr -1&0 1&2] , then find (A + 2B)’
- (i) a = [c 1 -4 3] b = [lll -1&2&1] For the matrices A and B, verify that (AB)’…
- b = [lll 1&5&7]a = [0 1 2] For the matrices A and B, verify that (AB)’ = B’A’,…
- If a = [ll cosalpha -sinalpha] , then verify that A’ A = I
- If a = [cc sinalpha -cosalpha] , then verify that A’ A = I
- Show that the matrix a = [ccc 1&-1&5 -1&2&1 5&1&3] is a symmetric matrix.…
- Show that the matrix a = [ccc 0&1&-1 -1&0&1 1&-1&0] is a skew symmetric matrix.…
- For the matrix a = [ll 1&5 6&7] , verify that (A + A’) is a symmetric matrix…
- For the matrix a = [ll 1&5 6&7] , verify that (A - A’) is a skew symmetric…
- Find and 1/2 (a-a^there there eξ sts) , when a = [ccc 0 -a&0 -b&-c&0]…
- [cc 3&5 1&-1] Express the following matrices as the sum of a symmetric and a…
- [ccc 6&-2&2 -2&3&-1 2&-1&3] Express the following matrices as the sum of a…
- [ccc 3&3&-1 -2&-2&1 -4&-5&2] Express the following matrices as the sum of a…
- [cc 1&5 -1&2] [cc 3&5 1&-1] Express the following matrices as the sum of a…
- If A, B are symmetric matrices of same order, then AB - BA is aA. Skew…
- If , and A + A’ = I, if the value of a isA. 3 pi /2 B. pi /3 C. pi D. 3 pi /2…
- Using elementary transformations, find the inverse of each of the matrices.…
- [ll 2&1 1&1] Using elementary transformations, find the inverse of each of the…
- [ll 1&3 2&7] Using elementary transformations, find the inverse of each of the…
- [ll 2&3 5&7] Using elementary transformations, find the inverse of each of the…
- [ll 2&1 7&4] Using elementary transformations, find the inverse of each of the…
- [ll 2&5 1&3] Using elementary transformations, find the inverse of each of the…
- [ll 3&1 5&2] Using elementary transformations, find the inverse of each of the…
- [ll 4&5 3&4] Using elementary transformations, find the inverse of each of the…
- [cc 3&10 2&7] Using elementary transformations, find the inverse of each of the…
- [cc 3&-1 -4&2] Using elementary transformations, find the inverse of each of…
- [ll 2&-6 1&-2] Using elementary transformations, find the inverse of each of…
- [cc 6&-3 -2&1] Using elementary transformations, find the inverse of each of…
- [cc 2&-1 -3&2] Using elementary transformations, find the inverse of each of…
- [ll 2&1 4&2] Using elementary transformations, find the inverse of each of the…
- [ccc 2&-3&3 2&2&3 3&-2&2] Using elementary transformations, find the inverse of…
- [ccc 1&3&-2 -3&0&-5 2&5&0] Using elementary transformations, find the inverse…
- [ccc 2&0&-1 5&1&0 0&1&3] Using elementary transformations, find the inverse of…
- Matrices A and B will be inverse of each other only ifA. AB = BA B. AB = BA = 0…
- Let, show that (aI + bA)n = anI + nan-1bA, where I is the identity matrix of…
- If a = [lll 1&1&1 1&1&1 1&1&1] , prove that .
- If a = [ll 3&-4 1&-1] then prove that a^n = [cc 1+2n&-4n n&1-2n] where n is any…
- If A and B are symmetric matrices, prove that AB - BA is a skew symmetric…
- Show that the matrix B’AB is symmetric or skew symmetric according as A is…
- Find the values of x, y, z if the matrix satisfy a = [ccc 0&2y x&-z x&-y] the…
- For what values of x: [lll 1&2&1] [lll 1&2&0 2&0&1 1&0&2] [0 2 x] = 0?…
- If a = [cc 3&1 -1&2] show that A^2 - 5A + 7I =0.
- Find x, if [rrr x&-5&-1] [lll 1&0&2 0&2&1 2&0&3] [x 4 1] = 0
- A manufacturer produces three products x, y, z which he sells in two markets.…
- Find the matrix X so that x [ccc 1&2&3 4&5&6] = [ccc -7&-8&-9 2&4&6]…
- If A and B are square matrices of the same order such that AB = BA, then prove…
- If a = [ll alpha & beta gamma & - alpha] is such that A � = I, thenA. 1 +…
- If the matrix A is both symmetric and skew symmetric, thenA. A is a diagonal…
- If A is square matrix such that A^2 = A, then (I + A)^3 7 A is equal toA. A B.…
Exercise 3.1
Question 1.In a matrix A =
![](data:image/png;base64,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)
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23.
Answer:(i) In the given matrix, the number of rows is 3 and the number of columns is 4.
Order of a matrix =No of rows × No of columns
Therefore, the order of the matrix is 3 × 4.
(ii) Since, the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.
(iii) a13 = 19, a21 = 35, a33 = -5, a24 = 12, a23 =
.
Question 2.If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Answer:It is known that if a matrix is of the order m × n, then it has mn elements.
Therefore, to find all the possible orders of a matrix having 24 elements, we had to find all the ordered pairs of natural numbers whose product is 24.
The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and (6, 4).
Therefore, the possible orders of a matrix having 24 elements are;
1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, 6 × 4
(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.
Therefore, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.
Question 3.If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Answer:It is known that if a matrix s of the order m × n, then it has mn elements.
Therefore, to find all the possible orders of a matrix having 18 elements, we had to find all the ordered pairs of natural numbers whose product is 18.
The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6) and (6, 3).
Therefore, the possible orders of a matrix having 24 elements are;
1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, 6 × 3
(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.
Therefore, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.
Question 4.Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
![](data:image/png;base64,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)
Answer:In general, a 2 × 2 matrix is given by A = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAF8AAABACAMAAABLEluJAAAAAXNSR0ICQMB9xQAAAH5QTFRFgYGBiIhmZoiIf39dXX9/f25ZWW5/f39AQH9/f25Ibm5ZSG5/gFszM1uARkZubls1NVtuW0hIbkYzM0ZubEYdHUZsW0gdHUhbbTMLWjMdHTNaRzMeWjMAADNaNB0dAB1OMgAySBwAABxIHQAyMwAAAAAzFAAUHQAAAAAdAAAALn3B+QAAACl0Uk5TAAwMGRklJSoqMTExTExWWFhjZGRzc39/jY2NmqGht7i7vLzK3d3l7Oygi8omAAABkElEQVR42u2Y21KDMBBAw8UqItbS1CuorUXI//+g0yRlQgi7WzF2RpOHPrSHM5Ap7VkYC+vMK9mLw+rynxJy6RObo/899XHaPPj/nT96EaJ7IAoQ2OGPts2C3YuCpkdgp38Xs5v+xsD8MOze/2jZEv0Y7PQv2+eS7Idhl/+py8n7g8EOf9bWjOzHYIdf0vqQynh1LhuOtqJJsfNv0otXUV8vGJcHcuBabLjMWbWL4f0v226d7Lu7/mMO7NUIZtlHSv59wP3jvRiYPPizNfPqX8Ve/bxgPv18w5Jb4/tbQ34u//+5IN1sEqusBoH981fwB3/wB//Z/ZefVz78j29/z39K/2Osw39K/6Os00/vf5Tt/d/uf5id3f8IO7f/MXZm/6Ms3P/Hmp8cAMz8V7A1AMD9r2t+egAwWA1bAwDS/7rmpwNxwOr0NwcA9PmAfJ8YoFpiqlC/rHmqX6W/OQCgflnzVL9K/1VM96uapwZ6MRoA8P4/1DzNr+DhAAD7dc3TBgAFWwPA7z1f8vz8MyxgfQGZYUFpbUKnuwAAAABJRU5ErkJggg==)
aij = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIwAAAA9CAMAAABIiHHcAAAAAXNSR0ICQMB9xQAAAJBQTFRFgYGBiIhmZoiIf39dXX9/f25Zf25Ibm5ZSG5/bltISFtugFszM1uARkZubls1NVtuW0hIbkYzM0ZubEYdHUZsRkYzWzMzHUhbMzNbMzNIHUlJWjMdHTNaRzMeWjMAADNaSB0dHR1IMgAySBwAABxIMh0AAB0yHQAyHR0AAB0dHQAdMwAAAAAzHQAAAAAdAAAAc3+CJAAAAC90Uk5TAAwMGRklMTExSkpMTFZYWGNkZHNzfX9/f4uMjY2aoaGpqbu8vMrKytnZ2d3d7OwUUuH9AAACOElEQVR42u2Zf1eCMBSGwekiIVMhs6hRUWYR7vt/u04wYEN2NmRXdjrev+bZnI9398c7cJyLXewc5qb0Z2oLzMpzyPvEHufgz6k9MOgNEAbPdVZde/X6NRyLr7l3HLDBEi5k8Kvm3u5L6ZswAExVtnd88ORrdn9R4hdZFEYOuoE6pCpP1TAoCxyHUErlK4c6JtFfDF1gcN4jAnwolzALu/cPo27yCBSGVI1m9vztyWCqOZwnkCxuymBwJoSlAIOqOZSBBk2zvZtKYeo5lIE2bJzX/5XIYao5EfgcMCGtLRgdZjzPcFGgBQMaM9z2GjDA2dQgoIwGahjQOlNXYJxTSiMFDHQF1utNrE7LelOr4bMmD9W1WbuWdW1tmKu7Pe/b2QelO17z+qoEIYfbQshwOuxk7fS150KhDA3Kh6ryF1Y53XmiDhuAw8G46f3EmYl3wt4a2BRMecloFVK81bxLylwoqo/jzxKYtog57d7UTkhRfTSKQwuml+7Vysh2NpWfuZYbSWGM35VJyxOkh2di07ewATB8qaenmxEY3/y1XQKjjhndRD7HMflPRSV4mBhKo3WlPthYCZM0Kxe56LaBOrGo5aX6QFxdl8GUzeiRrVywIzQnZTfTjjFRNd8NiHrlH2I1Y9UDApBHX+5yfjRuFIfOt3TtSPlottNacRjVo23lM551KZ/RrFP5jGzEnvcExpXPsMOy6S1BHNjDEkb2sPhriwJma5FfTCqfgWZW+QxkMa58LvYP7Bd0d07H2z2CGwAAAABJRU5ErkJggg==)
Therefore,
a11 = ![](data:image/png;base64,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)
a12 = ![](data:image/png;base64,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)
a21 = ![](data:image/png;base64,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)
a22 = ![](data:image/png;base64,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)
Therefore, the required matrix is A = ![](data:image/png;base64,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)
Question 5.Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
![](data:image/png;base64,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)
Answer:In general, a 2 × 2 matrix is given by A = ![](data:image/png;base64,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)
aij = ![](data:image/png;base64,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)
Therefore,
a11 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACsAAAA1BAMAAAAqrw8YAAAAAXNSR0ICQMB9xQAAACRQTFRFgYGBf25Ibm5ZgFszbEYdHUZsRkYzADNaHR0AAB0dHQAAAAAAv2kiNgAAAAt0Uk5TADExTHNzfaHZ2eyO7H23AAAAPklEQVR42mNgGESAKSMAm7BlN1ZhBuuRIVyATVRq9+6AwRRyLLtBIIBMQ7h37969gfpmExNyoyl2YFIs9QAAOnooOGhLtS0AAAAASUVORK5CYII=)
a12 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAA1CAMAAACUeyDVAAAAAXNSR0ICQMB9xQAAAGNQTFRFgYGBiIhmZoiIXX9/f25Zf25Ibm5ZSG5/SFtugFszM1uAbls1M0ZubEYdHUZsRkYzHUhbMzNbMzNIWjMdHTNaWjMAADNaSBwAMh0AAB0yHQAyHR0AAB0dAAAzHQAAAAAdAAAAnkdnLwAAACB0Uk5TAAwMGSUxMTFKTExYZHNzfX9/i42NoaG8ysrK2dnd7OwlcsEXAAAAe0lEQVR42t2SRw6AMAwEQy+ht9DZ/78SiSIsnAsnEHucjFZxYiHejZsOyQ0F44CEicFvWHMjzgxAY342oPnmFe0W6HzNMyMkyFCZKWyFySJatJ8sHiutqXc2NExzeqaJMmQo1uxGxMtyblXbhIV5IbnPRpdIHv+jmeNpVqwyCqaZrwIeAAAAAElFTkSuQmCC)
a21 = ![](data:image/png;base64,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)
a22 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC8AAAA1CAMAAADmtEJjAAAAAXNSR0ICQMB9xQAAAGNQTFRFgYGBiIhmZoiIXX9/f25Zf25Ibm5ZSG5/SFtugFszM1uAbls1M0ZubEYdHUZsRkYzHUhbMzNbMzNIWjMdHTNaWjMAADNaSBwAMh0AAB0yHQAyHR0AAB0dAAAzHQAAAAAdAAAAnkdnLwAAACB0Uk5TAAwMGSUxMTFKTExYZHNzfX9/i42NoaG8ysrK2dnd7OwlcsEXAAAAlElEQVR42u2UwQ6CMBBEC7RCoSAiAgro/P9XGm00nMzOhUDsnF+nu9vtKBUkkB6AaybGzR0v5UI86utY6R5zIrR3/tQjpZq4SP0/VXWUvblR9uqcU3hZUbh1XPEnzr19b0UTy/DC7wOELRSexo/3PRxHZh52GkHOb098ia+q9evBQjuqn/ndPtC4G0I+h3z+23zeqp5+DhJ/+2Pd0gAAAABJRU5ErkJggg==)
Therefore, the required matrix is A =
.
Question 6.Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
![](data:image/png;base64,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)
Answer:In general, a 2 × 2 matrix is given by A = ![](data:image/png;base64,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)
aij = ![](data:image/png;base64,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)
Therefore,
a11 = ![](data:image/png;base64,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)
a12 = ![](data:image/png;base64,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)
a21 = ![](data:image/png;base64,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)
a22 = ![](data:image/png;base64,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)
Therefore, the required matrix is A = ![](data:image/png;base64,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)
Question 7.Construct a 3 × 4 matrix, whose elements are given by:
![](data:image/png;base64,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)
Answer:In general 3 × 4 matrix is given by A = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore,
![](data:image/png;base64,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)
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a32 = ![](data:image/png;base64,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)
a13 = ![](data:image/png;base64,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)
a23 = ![](data:image/png;base64,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)
a33 = ![](data:image/png;base64,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)
a14 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASAAAAA1CAYAAADh7zYcAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAW3SURBVHja7Z09jttGFMcnvXMAGx51dr8a5QaBTCROuUC0grtgC2GhIg6gYuHQ2y3g+ABx51RJkTpFfAHvCRIgPsHuGaSI1IcpLT9myBFnhvoV/0YSKL73e/rz8XFIiaurK4EQQi5EEhBCGBBCCANCCCEMCCGEAa0+JMQXJKvDRQBfGDiKp/TN6XR0MhyqV1KKf+Xw4iWF0i3BFwau4yl84/VYPRe93j/JhpZutqBAuyX4wsCHePQ2SoF2+0cAXxjYj2eOASH4wgADQhQ/woAwIARfGGBAiOJHGBAGhOALg+4YEAvamqvtHJoW/zEw7hIDF7xcGtACE2lc/AvPi7/zjANgMNdeaWzwWQwIYUAwwIAwIIofA8KALBjQT40MKNnxTXKEOntfFYhNcMl3tZE4lzONvO9us/hN+YbK2JR9AAzumUpJLK0aUCaeueif/VbpGUVvxNPoRApxuw5gqzKXtgEuno6e9KW42X6n7N+MpvET24lyfRPgxVC+FEJ9ehHHD10Ufx2+NhkrKT5mGZ9OLp/6kP9AGGxNpaqO2zSgWp7hU+uaBKBU9Ob8+vrBrhnlF0qjFtHhTYBrUHcuDcgnxiszssu4bv4DYTDfdhoVdeziFMwoFl+K8/r6/MFQDV9tCnO/WNT49XPX595Wij9+8XDd4S2OzYBcMDbNf0gGpFPHGJClo5UvBpS07k0M60zJtzIax9Fj8eEYOyAbjJswqMo/BoQB7RWMeJ8M53LnOOOvvpfq7G3R0TZS6k00jU98MaDk+5L9TfcNA9JibJOBTv4xIAwo1WRy2k9b5UH0537LvmMyy0LaN6FtgVUUdZsGlLT+Azn4PSn4uga0uWpUpVAMKBmgVjG2xUA3/4EwsGZAruPxrjg/F0eauEWa2KXB6JpQkSm5NqCk9d90Y3UMaHXVZvfqQpEOOdOyxliKv3UZ22Cgm/+yGKeR/NETBlYMyId4DlqcTdw1KZL0FCu9WlFuEpuC6qln73TNR9eA9vd1U/wmR4j9H0yXTsEsMb4t49CUgUn+OQVzeAqmW0xFwLPgbB2xqy6ZZltsJcQn8Tj6oHskrTKgghgWa+28XjRATfd/aYjZvM1m519ufwCz2aNsLg9Z/KZ8fWDclIFp/gNh4I0BNY4lZ2e11daRIx1SlhjQTgck5R82O6Cm7X/JjzTzI1L/bWI75PzBlK9PjOsy0Mz/zvf6xiDHIL2ZATWOxdcB5b0iKuhscmdAmibk4jI8MyBzxrYZMAM6khlQnXau0GByEhDSVbCmBuTrDOjQjH0yIGZAHTagdEl+MsNJ7wua9JOkJYWSLhwr6GaarAOqcxMgBtQ+Ywyo3ICq6hgDMiiKZ6r3bqftk4O/ovH0G9tB170J0KoBpZehP89+jsGA0rgHvV+bMrZiQAX5D8mAdOoYA0KdyiHPA/LvFMzWZzEghAERIwa0PysxCBYDCrD4YRwcA68NyCiWojcyDwZbrZeQ/Zuqc3UMKKziz2M8HF18iwF5z8BLA6oVS/6GNitT7y/cKntkAgYUTvHD2BsGt7kMRvF3IRlQ7Vj2X9iuyYjGcfapdesbRBdlq1UxoDCKv5Dx6gbRo2fcGoNlvmsy8MqAtGKZzR5pGVD6yMzh9IeiohVC3pWsrZljIo2L/+A5hHH4DHwyoNJYUhOSd19PLvtGM6A8mdyvg8IUjGFwkFh0O6Cqtt3kyXUoLMEYBoc4NSu708CozZKif8ORsbuCsS8M1McuMNjGUtD9GBlQcr9OG/9agFy2yzCGgT2NlfylKhatDdm4/wn5LRjDwEUslRtK7rTNm3Cj7gjGMHAVS+U5nFLjmALp9swBxjCwHMvPup8vdbH9Z+kmupycPs3+tS4K+0gFYz8ZpM9OCoxBnVhyN5T9N4o8MSsIXzCGgQ+x5G6o/Pmw8rbs30ZRGMUCYxi0GYuVldAIIWRTJAEhhAEhhI5P/wNlrt1vO4lWGgAAAABJRU5ErkJggg==)
a24 = ![](data:image/png;base64,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)
a34 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATMAAAA1CAYAAAAtdgz3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAc3SURBVHja7V1BbttGFKX2yQFSZLSqs7dGukEhE2myNBKJ8C7wQjCIplloYaSMdwacXMCrppu2i27bRXIB+wQpEJ9AvkLtakiRohjOcIYckjPkE/A2EsAh/5v/+P/M/yPn7OzMAQAAsB0wAgAAEDMAAACIGQAAAMQMAACghJitPwMYqycTAlyDB8PskEUpMfP92f50St8S4vxLpidvYNzuAlyDB9MQBEePqON8XQvY/Rbk9ofF6UhJzN559JkzHH5hRmUXwQTvLsA1eDARJ1PyZm2Hux3Q+UdedCZnYBi2P84ErsGDKVHZ8ODyeLl8KJ2SwrAAuAYPpmFOnY+hDcj4n+ns5EeZNUQYFgDX4MGsqMx394njrDapZbRWRkbXR8vldxAzAE4EHqzDYrH43nMnP22EbS1q9EYkaDAsAK7Bg9E4Pz9+4BLnUyhoo/lv2AAAwDV4sDf1TMo0+NGZVsOi0E8vmranqhP1ge82ntFUn2ubb98lPzsOWSnXmZU07D1ESOvkuTfVifrCdxvPqOhzd02JTJNj5UZn4cYAvWoqMoOYQczAAcSsPrsQ9xOv9gxiBkeCmEHMrBAzj5L3rh/sc++vKEcOZvR5uIsgaCPQTbxsY6ltY6muRTTpSOweYieS4bouvlsWrkHbYqbKg06BKWrqbkrMmGgN6cHl4eL0CRuP7Way7yae/1J4/+L8NClcSyB6W+ggPvBne5Q4V8mYZHTNHqqeHLy5sXiI+s/ozVEQPGrLkcpwreP+lsvjh3NKPmzHJreT2cmsDSHj8dCkmJX0OS0Cs9ktvMk2dacjoabEzPcmL9J2YMI284O9QjE2KSRnZFLqXhyfnz+IxWZEnGues9syVsHkvW1bzNrgO6wdeux8dh67n2MO1pP4pYyINsmDBRxoERheU7dJaaY1YsYm95RO38YTOzvRqPfuma77bHIs0ZswEs9NZXPPxIwXCUXf8495aZqHPoiZbFM3xEzTW7MRgVEcizle2SgiTK9cLwijk56JWZLSZN78O1FSzm9t8NAHMZNt6oaYVbveIDQ0Z2KztITQ+QfeeoxL6YVo90NlLJ1OxBZ42X0nqVbfxEwgWHnpZ5s8dF3MVJq6IWYlRWyxOByF4f/Y/Ys3qeOerayghULGJqeEMMmOpcuJWFQyJuM/mNNUEbO8nSeVI4aNiMxyBKvIJpp5+L2Ih6JnbJsHXQIj09RdNFbrtjBNzFJNpcmbgomVrKDFQsaL2KqMpcOJWFoTR4tlxSx3sZaDOhfTy/K9fe5v18ZEQtcGD6JnjNpr2uVBd7S009SdKQ0RjWWELeqc3FWUmhk1TCM3bwqRAeLJyLZwZYWszFjZ+46dSPbNk3W6rqWZKnxvaqnu2LPv1BO5JIgdSTYCqJMH29JMHdERr6nbqjRT1hA8g6SJ1xU9FJUvVHmjq4zFeZ64Jmfn+7wNhPDaa5FN2y9JhzeTJqdI8b5GJxi0zbfvuU+3dX5kxeyzTnVes+vyNmE4EUBtPNQtZho4SARGZ3S0aeputM6ssi1yDCONpiKJaLeFL2bxZAwjM0L+VI3MVMYqm94InD3liPRretw618xUuW6C7zLrZao8CBw+l4e618xK8DBoYlE+erGPrjPzsdY1s/Xnv0q2MHEDIFcIOBFXdo0syvnLC5poLB1rNSrO29U1syLBVy3DqZOHvq2ZxUiaulN+0Os1s7JhZt5EyzNA8ltGuJJFTIl1F9mxTBIzU9fMqvAeC1lR/51JYmbimpku5DV1ozRDOqyd7YVrXmF/5GIUr2WERY2cKEtUZxZOUE6dWZmxIGZ6+Y5tztbNogp8spKtCYSY6RMYNudlm7ohZgoT6oAOL3fCUTL+2/X8p7ofWtdYWpwo3AbfXSvrupjtlMQMh18mrve67KZNnOLUxUPXxSy1i1/Y1A0xAzprT5xn1t00E2IGMYOYgQOImYliptqGADGz25HAt5UcdFbMlG3B++F0cfhkezRK1HzKOuohZt10JPBtLQedFLPt+YK7thDWqeVfKK6E/7aoUFQDBDGz05GEfM+C5xCzxjhYleCgc2JW2hbZL5JaK9cL0qewJs2ngupsiJl9jgS+DeGAHZYg4oD392odE7NKtshTRTr1X/EmfbZfK/uwECG9k6eJtyD4tpuDLomZ0BahoPFPIVYaSKVvEbAf4BscmASPOr8qRWZF6YjKSayAvQDf4MA4W2xaFLnHequEf9kueqC7AN+mcECvwEHKFpyoTEnMWA9XE38qApiS3oBvcGBSikneF9lC6kJVexABuwC+wYGNtii8EDvXKG93AegmwDc4sNUWhXkqpV4Ao/ZnXQJ8G8HBL7CFui2Eipg9K52BtVxQ6l5UObIFMPMNCL7Bgem2CM8i5Ngi90KbM45uU+ei7wC5fLcAvsGBgbZYqdoi90LiM7zLnwgKmDlxwDc4MMgWL4psoaUDAAAAwFTACAAAQMwAAABMwf8CDHtM1RGZHgAAAABJRU5ErkJggg==)
Therefore, required matrix is A = ![](data:image/png;base64,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)
Question 8.Construct a 3 × 4 matrix, whose elements are given by:
![](data:image/png;base64,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)
Answer:In general 3 × 4 matrix is given by A = ![](data:image/png;base64,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)
aij = 2i-j, i = 1,2,3 and j = 1,2,3,4
Therefore,
a11 = 2 × 1 - 1 = 2 - 1 = 1
a21 = 2 × 2 - 1 = 4 - 1 = 3
a31 = 2 × 3 - 1 = 6 - 1 = 5
a12 = 2 × 1 - 2 = 2 - 2 = 0
a22 = 2 × 2 - 2 = 4 - 2 = 2
a32 = 2 × 3 - 2 = 6 - 2 = 4
a13 = 2 × 1 - 3 = 2 - 3 = -1
a23 = 2 × 2 - 3 = 4 - 3 = 1
a33 = 2 × 3 - 3 = 6 - 3 = 3
a14 = 2 × 1 - 4 = 2 - 4 = -2
a24 = 2 × 2 - 4 = 4 - 4 = 0
a34 = 2 × 3 - 4 = 6 - 4 = 2
Therefore, required matrix is A = ![](data:image/png;base64,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)
Question 9.Find the values of x, y and z from the following equations:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
x = 1, y = 4 and z = 3
Question 10.Find the values of x, y and z from the following equations:
![](data:image/png;base64,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)
Answer:Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:![](data:image/png;base64,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)
x + y = 6, xy = 8, 5 + z = 5
Now, 5 + z = 5
⇒ z = 0
x+ y = 6
⇒ x = 6 - y
Also,
xy = 8
⇒ (6 - y)y = 8
⇒ 6y - y2= 8
⇒ y2 - 6y + 8 = 0
⇒ y2 - 4y - 2y + 8 = 0
⇒ y(y - 4) - 2(y - 4) = 0
⇒ (y - 2)(y - 4) = 0
Hence, y = 2 or y = 4
when y = 2
x = 6 - 2 = 4
when x = 4
x = 6 - 4 = 2
Question 11.Find the values of x, y and z from the following equations:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAABgCAMAAADM4avpAAAAAXNSR0ICQMB9xQAAAP9QTFRFgYGBAAAAiIhmZoiId3dmZnd3boBuf39dXX9/d3ddXXd3ampqf25ZWW5/VWp7f39AQH9/f25ISG5/WVl/e2pERGp7bltISFtubFlGRllsgFszM1uAf1kzM1l/bls1NVtuNVlsM0ZuW0g1NUhbbkgdgEAVbEYdHUZsbEYdHUZsRjNGW0gdbTMLWjMdHTNaHjRIHjNHMx5HWzMAADNbWjMAADNaNDQdNB00HR1JSB0dHR1IHR00AB1OSB0AAB1IMgAySBwAABxIHR0dABVAMx0AAB0zMh0AAB0yHQAyAB4eMwAAAAAzMwAAAAAzFAAUHQAAAAAdHQAAAAAdAAAAAAAAeeb0SQAAAFR0Uk5TAAEMDA0NGBkZGhokJSUmKioxMTEyMkpKS0tMTE1NWFhZZHFxcnJzc3R0fX+NjY2ZmpqgoKGhqKioqam3uLu7u7y8xMfJycrKytjc3N3d5evr7Oz+2pnXvQAAA5hJREFUeNrtm/tX0zAUx5t02DEF7cRXp6CILahDGRs+B50vdGyOkfz/f4un3aNJ2zS5GYPiSX7ZznKSfHbvzW3yPbeWZdoNb2jSljAnaEHuR2dAonZev0SmnXhKEuR25i/ID3FOqktygS+AKljQLwVUyo+lgKo0R+T8ebmgcHhsW5tkr1RQ/jD66l88EENN/JvvZNV+NkakUM6gE324444QyulTSgOnT9nNykwc9R/buEuP7dx+HEZ7mtJDZagpDQ57ttBSuEV7tuWdrQkscWcQWds5EfTj1/Y0StQtFdPgcFgVxxTu0meVT3VhXPi0kV4qvbB/UVePKdyNo2nKJgp0pz88anAxsrvHJf+OhRPoTL9lbYwDSKB7ZLiO8BvaKdx9Hu0l1ifzNgNtX9Tdz7a4f3VwCMtTD3+Q8/1t2iiG+kaDgm3tjgO/qL99VoU/ZnCYjMqDcj/e6iZJIzsxDk+P6mKop6Sh8ezzmVE5UKsfqpY7HtbEE7PuzfRzAaUKhXyyVfSYqYRRvt+lvXXhxDgMhFkbh70VhNDGC3UoVHn8izvCZKFa0XHG54866YmbdSHULPADQEogf17aCx9dnAO7bOcphATrXCOUT+lZtWxQ3ujn2o0/DnsdMNRsiIEyUAbKQBkoA2WgrhSKVxrLAeUMKKXJrbocUPHB/rBcllo9WAHGFFSP0oBq0y9PEARKpldl9Cg4lDsihJ7WIJaS6FUZPUoEhRBnU27B29sjylzJZ0Pu/r0nmlWmV6X0qHR79T3+2EnUmSBnQdxNlKDZkMJAl+hVWflAJ3k6/WEVtPskehWvR+lmdB+YpyR6VbvwcqoSU3G4/wZZSqJXpfQoAVQ6pjILesleUYCS6FWFAaXkvth2iFFy5FASvSqjR8Ghmu/XUaW5BclTEr0qo0fBoTZHhLxbM+cpA2WgDJSBMlAGykAZKAO1dKi0HFEGqFidopTOb6MK1/YF29uvuT8zC3qTU35yb58NuU6o5qP4WpTIFXKoRfUpKZRzf3LlnVeqyKHA9VJwS2XmlFsKWi+lCcV4TyWmgPVSmlCM91SggPVSmlDs/1TZfbB6KT0o1ntKULB6KT0o1ntKULB6KT0obkal5Amql9KC4rynBgWql9KCcvctKBSoXkoLqtUAQ4HqpXQzOgwKWC91iVDi8xS0XiozfhmlStB6qSuBWrT9f1C1Jb03I4SqCd+bmUP1r/q9mb78vRnTbm77BynYFoTTuMz+AAAAAElFTkSuQmCC)
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
x + y + z = 9...(1)
x + z = 5.....(2)
y + z = 7.......(3)
Putting the value of equation 2 in equation 1,
y + 5 = 9
⇒ y = 4
Then, putting the value of y in equation 3, we get,
4 + z = 7
⇒ z = 3
Therefore, x + z = 5
⇒ x = 2
Therefore, x =2, y = 4 and z = 3.
Question 12.Find the value of a, b, c and d from the equation:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
a –b = -1 …(1)
2a – b = 0 …(2)
2a + c = 5 …(3)
3c + d = 13 …(4)
From equation (2), we get:
b = 2a
Then, from eq. (1), we get,
a -2a = -1
⇒ a = 1
⇒ b = 2
Now, from eq. (3), we get:
2 × 1 + c = 5
⇒ c = 3
From (4), we get,
3 × 3 + d = 13
⇒ 9 + d = 13
⇒ d = 4
Therefore, a = 1, b = 2, c = 3 and d = 4.
Question 13.A = [aij]m × n is a square matrix, if
A. m < n
B. m > n
C. m = n
D. None of these
Answer:We know that if a given matrix is said to be square matrix if the number of rows is equal to the number of columns.
Therefore, A = [aij]m × n is a square matrix, if m = n.
Question 14.Which of the given values of x and y make the following pair of matrices equal
![](data:image/png;base64,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)
A. ![](data:image/png;base64,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)
B. Not possible to find
C. y = 7, ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:Now, ![](data:image/png;base64,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)
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
3x + 7 = 0
![](data:image/png;base64,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)
5 = y – 2
⇒ y = 7
y + 1 = 8
⇒ y = 7
And 2 – 3x = 4
![](data:image/png;base64,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)
Thus, on comparing the corresponding elements of the two matrices, we get different values of x, which is not possible.
Therefore, it is not possible to find the values of x and y for which the given matrices are equal.
Question 15.The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
A. 27
B. 18
C. 81
D. 512
Answer:The given matrix of the order 3 × 3 has 9 elements with each entry 0 or 1.
Now, each of the 9 elements can be filled in two possible ways.
Hence, the required number of possible matrices is 29 = 512.
In a matrix A =
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23.
Answer:
(i) In the given matrix, the number of rows is 3 and the number of columns is 4.
Order of a matrix =No of rows × No of columnsTherefore, the order of the matrix is 3 × 4.
(ii) Since, the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.
(iii) a13 = 19, a21 = 35, a33 = -5, a24 = 12, a23 = .
Question 2.
If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
Answer:
It is known that if a matrix is of the order m × n, then it has mn elements.
Therefore, to find all the possible orders of a matrix having 24 elements, we had to find all the ordered pairs of natural numbers whose product is 24.
The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and (6, 4).
Therefore, the possible orders of a matrix having 24 elements are;
1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, 6 × 4
(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.
Therefore, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.
Question 3.
If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Answer:
It is known that if a matrix s of the order m × n, then it has mn elements.
Therefore, to find all the possible orders of a matrix having 18 elements, we had to find all the ordered pairs of natural numbers whose product is 18.
The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6) and (6, 3).
Therefore, the possible orders of a matrix having 24 elements are;
1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, 6 × 3
(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.
Therefore, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.
Question 4.
Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
Answer:
In general, a 2 × 2 matrix is given by A =
aij =
Therefore,
a11 =
a12 =
a21 =
a22 =
Therefore, the required matrix is A =
Question 5.
Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
Answer:
In general, a 2 × 2 matrix is given by A =
aij =
Therefore,
a11 =
a12 =
a21 =
a22 =
Therefore, the required matrix is A = .
Question 6.
Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
Answer:
In general, a 2 × 2 matrix is given by A =
aij =
Therefore,
a11 =
a12 =
a21 =
a22 =
Therefore, the required matrix is A =
Question 7.
Construct a 3 × 4 matrix, whose elements are given by:
Answer:
In general 3 × 4 matrix is given by A =
Therefore,
a32 =
a13 =
a23 =
a33 =
a14 =
a24 =
a34 =
Therefore, required matrix is A =
Question 8.
Construct a 3 × 4 matrix, whose elements are given by:
Answer:
In general 3 × 4 matrix is given by A =
aij = 2i-j, i = 1,2,3 and j = 1,2,3,4
Therefore,
a11 = 2 × 1 - 1 = 2 - 1 = 1
a21 = 2 × 2 - 1 = 4 - 1 = 3
a31 = 2 × 3 - 1 = 6 - 1 = 5
a12 = 2 × 1 - 2 = 2 - 2 = 0
a22 = 2 × 2 - 2 = 4 - 2 = 2
a32 = 2 × 3 - 2 = 6 - 2 = 4
a13 = 2 × 1 - 3 = 2 - 3 = -1
a23 = 2 × 2 - 3 = 4 - 3 = 1
a33 = 2 × 3 - 3 = 6 - 3 = 3
a14 = 2 × 1 - 4 = 2 - 4 = -2
a24 = 2 × 2 - 4 = 4 - 4 = 0
a34 = 2 × 3 - 4 = 6 - 4 = 2
Therefore, required matrix is A =
Question 9.
Find the values of x, y and z from the following equations:
Answer:
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
x = 1, y = 4 and z = 3
Question 10.
Find the values of x, y and z from the following equations:
Answer:
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
x + y = 6, xy = 8, 5 + z = 5
Now, 5 + z = 5
⇒ z = 0
x+ y = 6
⇒ x = 6 - y
Also,
xy = 8
⇒ (6 - y)y = 8
⇒ 6y - y2= 8
⇒ y2 - 6y + 8 = 0
⇒ y2 - 4y - 2y + 8 = 0
⇒ y(y - 4) - 2(y - 4) = 0
⇒ (y - 2)(y - 4) = 0
Hence, y = 2 or y = 4
when y = 2
x = 6 - 2 = 4
when x = 4
x = 6 - 4 = 2
Question 11.
Find the values of x, y and z from the following equations:
Answer:
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
x + y + z = 9...(1)
x + z = 5.....(2)
y + z = 7.......(3)
Putting the value of equation 2 in equation 1,y + 5 = 9
⇒ y = 4
Then, putting the value of y in equation 3, we get,
4 + z = 7
⇒ z = 3
Therefore, x + z = 5
⇒ x = 2
Therefore, x =2, y = 4 and z = 3.
Question 12.
Find the value of a, b, c and d from the equation:
Answer:
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
a –b = -1 …(1)
2a – b = 0 …(2)
2a + c = 5 …(3)
3c + d = 13 …(4)
From equation (2), we get:
b = 2a
Then, from eq. (1), we get,
a -2a = -1
⇒ a = 1
⇒ b = 2
Now, from eq. (3), we get:
2 × 1 + c = 5
⇒ c = 3
From (4), we get,
3 × 3 + d = 13
⇒ 9 + d = 13
⇒ d = 4
Therefore, a = 1, b = 2, c = 3 and d = 4.
Question 13.
A = [aij]m × n is a square matrix, if
A. m < n
B. m > n
C. m = n
D. None of these
Answer:
We know that if a given matrix is said to be square matrix if the number of rows is equal to the number of columns.
Therefore, A = [aij]m × n is a square matrix, if m = n.
Question 14.
Which of the given values of x and y make the following pair of matrices equal
A.
B. Not possible to find
C. y = 7,
D.
Answer:
Now,
Since, the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding element, we have:
3x + 7 = 0
5 = y – 2
⇒ y = 7
y + 1 = 8
⇒ y = 7
And 2 – 3x = 4
Thus, on comparing the corresponding elements of the two matrices, we get different values of x, which is not possible.
Therefore, it is not possible to find the values of x and y for which the given matrices are equal.
Question 15.
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
A. 27
B. 18
C. 81
D. 512
Answer:
The given matrix of the order 3 × 3 has 9 elements with each entry 0 or 1.
Now, each of the 9 elements can be filled in two possible ways.
Hence, the required number of possible matrices is 29 = 512.
Exercise 3.2
Question 1.Le t ![](data:image/png;base64,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)
Find each of the following:
i) A + B
ii) A - B
iii) 3A - C
iv) AB
v) BA
Answer:i) A + B = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
ii) A - B
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iii) 3A - C
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QsUSDGcbeWEuzdkm4k6eUm41gmPgDvlFRSG2yOYK3HzcXWCZaLXKX7UgnGvov90eVtmQEhTm17rj9VkKmUYJeWdHdJz5f0zUCegyXhUOeLzt85bY5grcfNkZV7xgRjn8gK5M+JnfkMlgjY+PIUguEcp2jdNSRdxPZav7W90i1sjwIBUxU4lqeUYKugYP+EnDe3mSwHtjmCtR43R9aBYJ80SyK6JqD3/JJeJ+kDkZ06wSKBmroshWCY479o+w/2WcwGgyXtJJLeJunnZp3CZJ/TWhEMKyB7MowvEOwvOcLZfmRqDzbXXa1xM8U9dgajxjYfvkEnZ7Nl/hNsNl/q2wm2hNCK31MINo68f4CkFwT93tAMINeV9KFMmVoQDPK/wUztWDlLjl2kzGA1x82EUwTw/mLig4LbgqU0S1uso6uaEywXfTPdftVmo6cs9MMM9iVJFzNT9+eC67Gsfc1mMkz4Uw3HL87PqfNnXI+P6dZmSp6rDY1vC3L/NPK5HyzpjpKweub654ahUggWOy5WT3yHufGprCpY/qY0lsrcg4zPc4KlQJd2bcoMBik+YWbqSxqZxqNxcJOvISTE18Rxl7CdxaI/8EtNtRiCscd7iSQKmS+1m9iX+s6SfrV0ccTvsQRLGffc5gTPJRh4v2VCdvoD5ynj03XMl8l9fNBWNZ/BIl6MuUtSCEYfLLVuK2kVwYjswJyfY+youUS8ir08jwwiFgidyiVbDMFajJujYlYkWHzZc+K7HLdrSfpwpMvCCZaDvt2TSrC7mP+LjX4YczgsEbFOXT9TploEu4R9CB4f7D/4ot/XwrFyRFwiWKtxc2SFVOyLb2NL7nEft7L/Pd2c8j6D5SAccU8qwQi7+bpFa4ThSATXfkwSy7HXRIw9dUkNgrGnuZmkF05s7i9gbgYiL3LaKoK1HDdHVmINWdIPsaLjPrAqEqiNP5M9uBMsB+GIe1IJRpePk0TQK/F3wzKQ9f4rzVLHb0Rt57RSghFz916LvcM8HbbTS8IA89Ec4SQRscHsTTzm2BrZetwccdnv3sMCncduE8z0GKjwhWHGX2q+RFxCaMXvOQQ7mSTi+SDDU21/w7LrwqZQzMK5rZRgr5d0+xWDY3jBN/X9BAFxSVzQnvdSFg3xPUnfsFhMPjgYXWqPmyDi7KWY4TlaQ3jYdyWxhOWIyrsksTyc+giFnTnBCjSRQ7BhOM5GscbHukiEx5eD4N8csUoJljPmrt9D9AYuCtwrP7RZeHzUZun5nWBLCFWewQqGW7zVCbYI0dovcIIVQF4ygxUMO3srfjCWLhya5JS0t80jsLMEY+l1zhVRDzHQs7klp91cbOC2ESzmmfya9SKwswTDr8TDEdOW2yAWCSMJYZpqTrBcZPfnvp0l2DpU6ARbB8p9j+EEK9CfE6wAvD251QlWoGgnWAF4e3KrE6xA0U6wAvD25NadJRiOQY57E96T2zjST1maqXg0+nSC5SK7P/ftLMEw03MitcSKSCgMBxPdTL8/hKj9pDtLsNpAuZl+HYju3hhOsAKd+hKxALw9udUJVqBoJ1gBeB3dynGaX2bK6wTLBK7EyEG+doopkCKbIyCns+QpVDHZZGPfSo4Jjs6QZPO0VrSBA5YfqSzYtlaXCR+TIytUhOGc3lSelCVYnGBLCK34PWcG4wj6k+wIxEAocnS80Y6nz4VlFYgZdSuHPh9kp3jJtDQ0TjeT3IX0AZxfq9FaVZepIdu4D46qkEKPjGBXdILVhne5v1SCcZL305JIJBOWAXqOZZAlL2JuYs9lieevYNbiiPybJD1k9DIRoU/mJbI3ceDwByWD2Glm6qC1qC5TKNoJbr+bfVg4fe0Eq41uRH+pBINU95TEEjH0rQ2JR1HkuKxRhBhVLiFVHIl4qCTCjErJIRpLRf5/GUlkfKJ4RW5rXV0mV66p+8CANNmcyD6NE6wmtPF9pRCMVAHUlYJczAZhSRxStZHr4YmSSMu87sbLT4JR/H7jzMIUp+BjcCabfWITlk7J37q6TC3MmLUhFqsK0rM5wWohm9hPCsHGqbOnCMbyiwxFlCtlNtuWxozKspYSsiwdS1rr6jIlso3v5UPDcpnlMOkcnGC1kE3sJ4VgWOgI3eKFnSIYM9uRNsuReSkmoUqiuMmXE2ZGJqhvWz7E3GxXDLyO6jLJDzhxA0l9sBxidKI5wWqgmtlHCsEYguP8GDhYKoUVJNn0Y60jbyL7nRr1wnIei9AycjOSp5AEm9QDe0amBW08/jqqy+Q87/geEquS+59884OhyQlWimrB/akEI/88JnBSf42rq/B1x3T/WDs9DcFyUmcXPMrkrey/qCTCrItFbS7oOWbcdVSXiZFj1TWkNcdiSi3moTnBSlEtuD+VYAxFXnM2z+Tbw+cFuZgpyPeOcYH9Dn6iqQDjTVQSwReEcYZoBpau5AfMaTWqy7R8/gPsQ0JuyHFzguVou9I9OQRjaFJos5Fmn0X2JzLlYsXDyEFZHAg31VpVElmCAwMHTmiyD1MXK6fVqC7T6vnBniLvPGe4cnCC5Wi70j25BJsaHgczhR9KctOXPBazJmV58NWRSWvc7i3pxZKOsv1jriO8dXWZ3OdnZmbZTmLRsIEJezOsu6wqsISG9atXjeuhUrlayTxwSc5zYhDDlxT/FwUFyIYVU7urQOwT3Do2QEyZ40ntTTEIlopXLjB4tK4uUxMT+hqc7PztkRy10Y3oL3UGw3qIqZ4vOTPV0OgHJzMm8ZiCAhGiJV2C5RC5IPfBlot93MFLJR1iX2+soLmtdXWZXLnm7nOC1UY0sb9UgrEMO9zi27DODY31P0sRAms3lZGXfd+pJB0aYEAEB/455CJ9QkmNZrpuWV0mUX2Ll6NfrIqsNi6badn1JeIizPMXpBKMtTxLQZRGTB/HQSiRw0uMEWGI/ysQKftWrJks4Q6UdJhZOMmMTKVH5L5rYlWVOUFaVpfJfvjgRowpfAhwtBMAQMO98jtzpywVPh935wQr0EoqwRgKaxqb6mEvg2Hj6AIZat/KLIbBhdAtXqgj7INA8fSarUV1mZry1erLCVaAZA7BCobzWztEwAlWoDQnWAF4e3KrE6xA0QPBqHj4oqAfllT4k7YhaLfgEf3WSASIeMEFEzb8hwRJU0Skm8aGfBvaOL4ulOcYM3unVEPchmdyGfIQwMLKfnWqUYLWCZaHq9/lCOweAtsyg+0esv5EjoBFoDsQjoAj0AgBn8EaAevdOgIg4ATz98ARaIiAE6whuN61I/BPU0ai62qoqeAAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
iv) AB
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)
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v) BA
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)
Question 2.Let ![](data:image/png;base64,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)
Find each of the following:
A – B
Answer:A – B = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 3.Let ![](data:image/png;base64,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)
Find each of the following:
3A – C
Answer:3A – C = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Question 4.Let ![](data:image/png;base64,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)
Find each of the following:
AB
Answer:AB ![](data:image/png;base64,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)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 5.Let ![](data:image/png;base64,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)
Find each of the following:
BA
Answer:BA = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 6.Compute the following:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.Compute the following:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
.
![](data:image/png;base64,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)
Question 8.Compute the following:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 9.Compute the following:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Question 10.Compute the indicated products.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK8AAABCCAQAAACLQU6iAAADm0lEQVR42u2cTUgVURiGn+tfJUREgaBh6iIoDALbhBBEYUHRJqorCEWbCEpIpAglXBrRqkU/ULuiAgmMpEgMaiFCSkG7IIKCQDdthHYTcsdJwnvn+86Zc+cY553NDCjn4YH55u/lQkjI2k6BVtqSbTu1VSdoXLF+G01rlLFMNvCFKNl+sqXq6PtXrB8xtkYZK+h9Rl+8FVlfdfSWZPU+Zsrq9Z2xgt4hb0bVWFm9vjMGvUFv0Bv0+qn3KMOM0uu53nRKL/UWOMQVIs57rVdC6e1wWLp73OP9cEij9FbvMF/Z6L3eNEpP9RaYZNz7S1s6pad6t7LAJe/1plM61FvLcQY4SwN16hcrB4jYyz766WdzhnptmEwonentYJIiNTQzyDtGlOAjzDNEN9DFhzJvnfR67ZhMKB3pbecbxXj/DFHKHey/qWGKH7TH8+0tNzLRa8dkRulEbz0veE9dfHSCRTrUM60/OXrJ9Konsk6vLZMZpRO9PUScS47uMke9eqZ1JVpmmV31/3V6NUxtjHKjwnZSTOlE71N+syNZeI476rvJ72yK95v4xSQFa70aptYUvafElA70ruMzn2hILieLMY48E7xK9o8QMWg9e+2ZzCgd6F36q+fJ0el4ynVTI15nitFk/z4LNFvrtWcyo3Sgd+nUe5A81zziIw3Upj4krMx4skoL81zM4L7XnsmM0snsvcbreA71MsMbYBfHFOBFHsdSnnC77O2/bvbaMplROtHbyD0GOMhVuuhkmh6uqz4p1nOLyxzmIRcqPFvp9NoymVE6e2rbxs540UY6lTdmpRNud3Ipyuqh2JZJTxk+BuXAGPQGvUFv0Bv0Br1Bb9Ab9Aa9Qa9hJMUmt3qzJvBIr6zY5FJv9gSeDYf0YpPr4ZAtgWd604tNrvVmS+CVXkmxya3erAm80ispNrnVmzWBV3olxaa/Sa8r6fXqCEyYctQrKTYtR1JX0uvVEJgx5aZXVmwqRVZX0urVEJgy5aZXVmxCUVfS6pUTmDM50JtdZagUaV1Jq1dOYM7kQG92laFSpHUlrV45gTlTbsNBVmzS1JW0eqUENky56ZUVmzR1Ja1eKYENU256ZcUmTV1Jq1dKYMOUm15ZsQlFXUmrV05gzpSbXlmxCUVdSatXTmDOlOsbs/Ri03IkdSWTN2ZyAjOmXPVW43VJ+NYW9Aa9Qe//qPemN78RNlFWr++MFfSG3zGzZwwJCQkJCQkJscgfaUy/FXGAmgUAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 11.Compute the indicated products.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 12.Compute the indicated products.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKkAAABACAYAAABhnldlAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAATHSURBVHja7Z0/T9tAGMavc2FjacVlg4GNXNgZqshqoRtSk4ityhDRDICUgVLDFrVN927lA3StKsEXgE9ApTJ2gs8AjU2g4Dr22ffeH9sP0rOh2O9zv9j33tlP2OHhIYMglwUTIEAKQaSQjv+eTFMlzJCsveo+mfT6EaS+v/lMMHYx/sfr/8Uvvb6/XGbTtpp8J752ds2bWzvwiU59j2/LeB0LaZ3Vjzd9/xmMfAxvFFL4ZMZrQApIqwNpEeZhKvNGSkhteGVrzpznuOSQ9nob9WZT7HPOfsV9sCva620s1jk7Gxt1E4rXz5qtrVcmIbXllWrteeX3WwuCs9P7ueb4uBu9vUWjkB50xBqr1c5rNXYeFO8qpHu9F+MxYleRyXk4YKLlr5uA1JZXk9ovVWrPB6i33Gh4H7vD4cxjYMVFmmdabvfhADgK6XDYnfHm2Qn3Ov5DwzzOjm8HK900ytu9Sa+oas91XOG9vzvmQ3CDi4XoHKwB0ogxotl/O20AGeNXMstFRYSUqnbK8wGkGdUW7KjMV1Kq2qkaqPCYovPNSndfREjvryYSppUN0qy1q8IZNIxh49bwvncHg1lAmunWUz+TrbNMkN7WLk51X0XDL8Pt/Pe+YeP19pc0UEsBadKeueyaXFvwUdrcqKyQZq2dAtZ+Z+XNZJUhqH231JAm7a3L7P8mmVAFSPPUTt04pc2FK3+7D841ruOtAqR5azfdsFUa0nBJRnT8uOlD2SGd1P4hT+3kd8J57yS6hgpIg7paYp2L9ig6bw22DIXwPiWZVnRIVWtX6RviVhXSaieHNDiRwIRwB0O0j1x80GSrtdKabIvexEkGGApIbXhFUXv2q3ZrIXzWNtyr79WDOgeD7mzQsHHR+Wx0CWoyEb6UbVhsKOgqkxssuQeUVSG14RVV7bmWnhq1r4/q5I0fXqf/UqW5U7rdV0F4ntT+CgQgBaSAFJBCgFSzwq583OzAJ/NeA1JACkgBKQRIASkghQApIAWkECAFpIAUkMInQApIASkgBaSAFJBCgBSQAlIIkAJSQArph9S1vHgd50MBqU2fbB+bGNLVpTk292fV95dkPtBW9uU05c3ETNM78XTE+OufRfRJlydpypvHGvVaCVKK3E/awQhe2/339mOWTEydkNrKCL3zJG9OqPKdJ2ceKxmktrIvk86nKZr7eTMxdUFq0yfVnFAyWG1B6lr2ZdJ52oTURZ8qA2mSTGdfJk3YZTMxdc9JXfGJ0pPCQmoy+zJpILJmYpqG1LRPOjyxCKna+l/W3E8t87BoJqZoj1SjZKjfFjWVETrVkyAnVEO8DgWk2t8WNZ19mTQwkUzMHZcgteETtSeFhNRm9mVak6A696OE1LZPVJ4UDlIXsi91NiiUqXou+GSqaXMGUpXcTxOSycQ0AakrGaFUnhQGUhvZl0kdbHTAg8hBmUxM3ZDa8kklJ7QUkNrIvpx+lVLLxNQJqS2fdHsi8wXJk8dKBqmt7MvEZRaFTExdkNr0SbcnEs1ZrjxWBJY5sgQFAVJACkhhHHwCpIAUXgNSQFoGSAeD51l/r7Msiqt7bNxuLKQV9kmj1+mQCsZ+u7CsZOtbLPNbpVX3iUJ9j29n+V1YmAa5f9WFCRAghSBF/QU0jMr+PwOicAAAAABJRU5ErkJggg==)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 13.Compute the indicated products.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
.
.
![](data:image/png;base64,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)
Question 14.Compute the indicated products.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
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)
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAABgCAYAAADfChZeAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAVSSURBVHja7Z2/b9QwFMfNDGwwgPBNwMDGOd07oMPi11aJa9QNdTgdkShINxQUulVA2WGCjYUVkMqOyl8AUhmZyt9AiZ0ml4bc4ZD4R3zf4St16ru8j2M/O37vka2tLQItjuAEAIcAXPWfEXJiluBszSAV/d6awTheO8cI+ZEY+f236AGP4qsAo0cRpxvVfie/6WD8UBvwPunvrsXxOUBwQ+MBfdgp4C4tBbaWqCZ2OwN8NFrpDwbsCaXke/kH29DmaOVyn5KvicMPpWj/62A4vqnbbhwNLzFK9vIpOrG7Mtq87BXwpyG7RXq9b70e+Sacaxv45uhawpr8Kq2NEjwbxrf1weZXg4A/W9/ePnUcPvuh6uNOTekSvGXg29vrp/gF8pnyMC46nlOym0JXd35tu4w/zmwWB4EYfCx8egvANb1lbBDdmzUQkt3HL5O7DwC3qFVG3up6w2cFb9ImC994GaW7DDx/w2s4vwloEcTKoDHg79cnk9MAbmGqp4Tt6X675cBK44U8WKT91ZfltR3AtU/ndEd1HW0LfBQu3U3W74M6PgHwFlTlRNNBm2rsAOAt/KaqqN3VYBHAm27RWBhXBVWmZxhygX9WWccB/H+fbchuU7a6Uz7XFkeujPHnqkFU3ei8PJjygyDf1nDxoMLJ8jSLrb61+RFlPFwaHh2tHlZJx4CUx6jiU7M8Ox/1xfNPJuunRbBIWfjCq23ZUWBy8K/vuiYkIuNZ35l1fueXb3LQe3Xs+WnwgYfRjaYBprNTOqRnRwHgAA7gAA7gAA7g7ktubZNdDoADOIADOIADOIADOIADOIADOATgEIBDAA4BOATgkCfAXS4VYjNHXNV2Z4C7lh9elrw5aiivrFBKpXCXTi2JsRPAXcsPL2uaL24GeDq4SnfpFHPaOjWlu5iIIJ8xrwShH/hksnae9a6/rpNACOAtSl4T5mGc5obrB55mmSQ+oMFHUWKkbtwA4A1/j0hGmBYD0Au84rq2rC2zNpmcbw34vMJ6/4oQvc48SZ4toME78WymgE8D2NHFkC89yDJHpV1F6HOBVwYHisXeUqcsXzlDzvxcjuMrvgEXU3kWFZsGnqmQK35YhjjTh8tnN8jZ5S+Y0mtGycXfYAt4YYu2r2oba/j/rKOlJEKR45UDT6ZWK9mjbe3DsYYr7IELddrSv9m+yTc9DeaS4K3pG958DV+MChA2p/TcL5TvqqQoY0rvEPCq2TQt3TUNIL0B7lJ+uC3gAmyPXX8taqsW88OXwuiuVwcvLuWHz98e6V27C1Wb5PP3Av5qGMWXvD1pg1paFgEcwAEcwAEcwAEcwAEcwAEcwCEAhwAcAnAIwCEAhwAcAnAAB3AAB3AAdwu4rR7eLtleGOC2enjbtj0eDm7mPcQTu9dGm31HgLebanTc4XZ6eNu2nV7kpAdZr5Mo5DfEPTdVu/fZyR1C73zqFHBbPbxt284qP5Qvb9apPtFJ4DZ7eNu0PSulKGtHqXKLt5PA58l0D29TtvMBVdGBMK/5opBB6hVwkz28TdvOoVYAnzcYvAZeJ6lOj219/cPzLsIVb3GdzBdngDfJUp1OqWZ7eJu07RXwplmq2f+wlXpkwva8dXredO/llG6zh7cp2wjaitskSz28Tduetd/OpnuVJaXTwG308LZpO8uiTZaPRxUDYd/bgxf5kBZ6eLtgO+J0QxytivNzMcCyFGLVgLGTwG318LZtu/gb8jxxyvbq5Ih7d9IGAThkDjhuvLgmtJMGcAAHcAAH8FrAj+qP2rz0t4iq8rl24GmlXzv71UVWekCj9gEKDlu0mQBOAHDIY/0BaLtyUoOy+UsAAAAASUVORK5CYII=)
Question 15.Compute the indicated products.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL0AAABgCAYAAABId3PBAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAdESURBVHja7Z0xb9s4GIZ58zVbO7QovSVDt5rOniEwhKa9LcDJQraDB8PnoT1Ag9sq2YJe073b9Qd0LQ5o90P6C3pAMt7k/IbkTMqWbZm2aIqUaOkd3q0tSPoR+33k+30kZ2dnBILqJCwCBOghCNBD2/WDEvLTKtV17oC+woqik4eMkOvxD327LDryBtHTqs594NGX8nmTW9ruvwL0FYa+SZpfT6LoIdYjVr9NXwF6QL82JNiW0AXQQ1rQD3vHe01Kvo8huhOize9tv3/k3LwG/i6j5DIJWcbjPO4N9wA9tBH0w97hmHcySsW/An7mRy/cAd572mp577rn5/cWPwB2rTJPQA/ohc7Pu/e8x+Qb9YJoHiaPkq8x+GpA2ZYYJ/NeT8c4/yGMP9gbFpw+B/SAXgl6Dg1rD35b9TEQQm9cPumZQD8C9JCR05sOI59c2elXJbRijCz4CzE9lBv6ZKdXBKpo2Hu946ZIvFve524Y7gB6KDf0cdjALl3b5cXHGOcbScJNm50PKuADekCfEdrQC5U42cSurSIZ/INg/9fJqdPdGOY/AD2g14ZeBocNrbMLZNkH0omsSu4B6AG9VKcBey47zXFZqgk3oAf00l2TseCtLARxea4cZvLY+5Y+wwf0gH79n/fZC8o6F+lYmtsTGPP+zAKqCMni++RyTSEcy4S+zl5sl6X6m2wCfd/f9/mtZuK7SamI+D77fyF/lxFyFXttek0+5zDs7vCEm7LgvW6usrBgdfViu6xNfOKq0PMTkPVJpBu/t9jRW42PC3OmrS9eMHiWJ0E3dtwFFR/T6kJf93XCggF6QJ93waqQCxQ5bpPFEYC+BOjzmP1dEPd2tNvsDaXk3yISOdPFEYC+YOjj897ZMdemZv+yxS9oSKPxo9EgP4o4vbBRHAHoV/yurPPJOPQ8y26z9ps8Zn+nFsky9LaKIwB9gdCv280AvZ31AvQOQr+p2b/u0JsojgD0JUGva/avK/QmiyMAfQnQS83+rHPhgofDRehXFkcorBegdyy8kZj9Cw8VdIoUygpvdNYL0DueyBZ5bCkspzmKFFxIZHV94oDeAei5XK+udwn6TdYL0DsMvarZH9Bvtl6A3gHoZXEy9z6rmv3rBr2N4ghAXyD0E7P/dR6zvwvi4+bVRKKQYrxQtoxnJtYL0JcMvQmzf9maVdirVeTnPa2xURwB6B2J6SF7sb9J6F2ygZu0pgN6QL+kou3UmfPy20eJ3Xoc/h32hk1ADxmDvmg7tdohAh1Nw7xB4D0T3Ylz9MsH9BWPVfM0eyob+mljgvQY4stGdn0Sho8APaCvFPQx3Mt98ae31LpjA/SA3knokxbhkku5pDVNaq6AHtBvNfQJ2BLo130QgB7Qj3/Dgyf3yf3/DqLoyVZBPzXaSXbz2ZNAev6t04MHL8mDg38APXb6+kCPnR7QOx3eSKBfF/oAekCPRBbQA/ptgZ5r1Xl83o4agB7QOwv91PCXfkdq8jFc6XIJ6AH9koqyU6sobktOR9xvw8cxrRvO0zcJ0AN66e5ahJ1aGfxZgfwtoezSH0S7ptcJ0CO8qd06YcEAPaC3vWAuFCu41D9ftz89oDcCvd4VtqpcKVbgRQqihZ6hIoU84q/4TcYSP3A2Hk/b7x+p/N3f2c8XhP7yN6B3FHpXihXiUwp6kypSuMlTpKAP/GFTkkQK+FXGA+i3YKdPBlBW97CsIoUCQUlafXhBNP8ow6Sv5Z3KeAA9oM8OaywVKWh9gPzVFsmz9DNz1fI4AT2g19pZbXg7TEu1rR+gB/RKoY2NIgUrH6fCwwyA3gj09hesNOgt+rXNj5Nd4qE1B6HX7f8O6LNCG3qh6jMB9AVCn6f/e+nhjYUiBZOJ9ibrAugR3mx1IsvXRXaaA+gBfe6dVBbClP3sZ/zodPBWFkICekBvJK5fcTl1VQYo/IaYP0SXzoW4PWH+9XVAv6XQu1CsMLmgMlqkoD0Wf9+P35ea+G5SytoYAL3j0LtUrGC6SEF3DOsPAego60YW0G9JeAPZPe3RLRd0yWqtmtMAekCv9RuKZ4CmveAnVuvj3nCvzHmZtqAD+spDr24l4SFnq+W9m3d5xh9AeZd0NizoMrsGoK8h9OLegnmvl+4sSj6+XYAf0EOmd/o1Bw41hD4MH6n4aKBik7q0TEPP/01ha1ZweFYKekbIle4RGmRHcQMkNW+TDvQcdp48ilrdlve5G4Y7tYEe2n5tCr2I7eMSxaRGlzY7H8quKwD0kDXo5+Gfu7Rz5JVBQA9ZhD6dyJZdWwDoocKg51Kt0QX0UGWgF6a8kgtqAD1kBXrZUXTSjwcxPVQ16IXlgFeLCa9NT9isw7C7w2t0KQvelzkH0xZ0QA/oZzt6q/Fx4cyftr5M2x2WNn4LFnRAD+hrJ0AP6AE9oK8a9DANpiXzKGFhAD2ghwA9oIcAPaCHnIe+pjURqnUHgKVi0Ne1JmKTugPAAtVOWAQI0ENQ1fU/uHWfA8zQj2AAAAAASUVORK5CYII=)
Answer:![](data:image/png;base64,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)
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)
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Question 16.If
then compute (A + B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.
Answer:Now, A+ B ![](data:image/png;base64,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)
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B – C ![](data:image/png;base64,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)
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A + (B –C) ![](data:image/png;base64,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)
.
![](data:image/png;base64,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)
+ B) – C = ![](data:image/png;base64,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)
.
![](data:image/png;base64,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)
Therefore, A + (B –C) = (A + B) – C
Question 17.If
then compute 3A – 5B.
Answer:3A – 5B ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 18.Simplify ![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
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)
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![](data:image/png;base64,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)
Question 19.Find X and Y, if
![](data:image/png;base64,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)
Answer:Now, X + Y =
…(1)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALsAAABACAYAAABCxQawAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAbMSURBVHja7Z0/b9tGGMbPXpN0yhI3l80ZsoUn7RkCga6TbEJBEZ5caBBUAUkMcHAR2pvRJt6zxZncpWNR1PkCzleIPXax/RViV3f8Y4o6kkcej5LsZ3gAWzLFo/jj8Xnfe/ma7O7uEgi6DcKXAAF2CALst/nLImQ5Swt4LEs5x7ME2G+xfH/jASPkdAzC92nRM3vkP12k4xnZ9I38WMgl7QzfAvZbDrtFrKMN339wk49z2KFvATtgB+yAHbAr+uJGfXC476Wy+wbsUGnYtwfdx4yS48gHE2p97Q62HzcF+tDprMf7p+z4+WDbUoEesEOlYPdH9tNWy/69v7d3N/jdWQ3AY6dN2CDfYS/HQfO57Y5+4r+/du01Ssg5c/yXgB2qDfa9vf5dm9m/RaAnL4AxcBfM3Xlhcpyet7HCs0ZpYIPsy/hi87wVwA4ZDVC3B88tMbsahp3DOp7VL9Kp0OhiKwIZsENasHOv3GPkgDD3k8kxirvKQ/KFPLS/TN1ZonUC1jsA7JCRbMxg0LXGU/pXwuy/+p53z/QYBdAS2PMuBMAOacEu4KLkKM7GEHJFrd6+SeAjqyKbvWPYC4JkwA5VtjEcspHb/pn7dQF8Z7gF2AH7jQ5QQxDPTaYf83x5nsUB7FCtsHOJINUg7LF1yg1Q84NkwA7VArtIC1L7KG9m1VVWPl01zw/YoVKwy+pRPK9/j3tmao/eGB1naJeSsYEoH+jQrfFFcFI0fsAOKcMuSgMIOQlqYQa8HmWZg+4y+p4y930TYw1md3oe1sMsD522o7qgBdihcuUCrUcfkw9EUNr6O6pTaUphBugsKgRzRv6qqtWaJexRtaaJbdIb5D6iJfsbwwdd+MhYU4+UoZ69GdBHHfZL2x5tFvAwAffIbm+2naFTdO7jH1xGPkWLHwldRtF7YpHkMn7P4ONo4WNj6fF8Zxs766kMw8nEmAwFgIDdrLjVW6P0kFu99Pnj5dJiFTriYGwRO85wPYKbs7lGyb8iC5WzaDftN4Ma6CteTCQrCQ3KR8mlbFD1B1vOaniBiQOUgcYPNEjp8ZLW12vznI0B7Nkzeo/RD7LgPSqgSz4jG0Gf5lNM2Kx3kDXDyz1nBJjV+5zcsKmMQtZ4ZBefGNOj1kdVTwrY87XjshdFxWIGLrAt2R05OvfUdt8lnw24dhiTaxbx32esUmee2NAexIA1Vbm3m5NOk+WP+YxgumwWsJu0L0H9vewc8vMu8+/XpQ/Tpcxi/Bl1+9kHvcHW46tnvCG3L5SuHZq2LlkKY4qJ2hJxi2toTIDdnG2qsqqctRotW2cohD0JGLGsoxZtHc7yRF/fbegFzx8L+0Lpn6oBckFToMKMDmA3ZFEVyo4zt5O4jLzPVPtQw9V6JTM0IpYQgVSJ2CGnKdCUZJ8L2I1NYKdl9xcErew461xkzfqFs2Fwm7m2M2UibJ2ZtChYLTsbwMbMIew5JclFmZu8OC3Mzk35+ULQeYrx2s70Pjc1k9b5BQH2mwF7zGQBL6VhFwFpWMuR9suzPBlVYYdnX3zY+dhk2Rkt2GVZjtgvF6xSzSvs8OyL69n55MOZZMx9J3uvkmePPlSWzstbbFoE2GFjzMAua7GX1XYv/XrewybJbYTLsHr76Tuw6LrG7D+S2xZmY/gHev3+Ss+i+5FVkQ1U7JSDFs7wvJ1bk9DzAwzLFa740/rdfv+HpvYP2OUKExhXyRID2WtZr4flyFJ7HLTxE+XJF6kaqVjpfUR3C1nJQ3CAwQJS7q08DFKnbvnJwqwGZvQzneAWsNcPe2QPk+dB9lrW61E8KEtt8zLlfMs5XYgYTIbyh1Ru7EkD7PPv2ScughpiwaLaLYAM2GcOe+wccioWVWKHomd8AbJB2Kv2SK8pvqnUG35WsMfBKoe15AzPZ3RRIlxQJwWQlWF/9uQ+uf/fM99/ojzTVOyRriud3vC/sjsfCH31z6y+Z+7TVXLpkzaovamyDUA2BLtOj3TdQF6nN/ysYTd6twPI9cOu2yNdywpo9oYH7FAp2HV7pNdva9R7wwN2SBn2Onqk1x2olnnCDLBDyrDX0SO9Lsir9IYH7FCJjmD6baNr8e4Ve8OjIxi0ULAn91e2Nzxgh9Rhr6FHeu1jL9EbHrBD5Xo9avZINyHV3vCAHSpVLqDbI92EVHvDA3ao9H+41umRrnVCNXvDA3aodCGYTo/06t5cvzc8YIcqVT1W7ZGuk33R7Q0P2CF08QXsgB2wA3bADtgBO2AH7IAdsAN2wD5fsHc970dZy7yFO/EZ7QD5WgBgB+x8qf+bav+SeVdeO8Am/40QYIcgwA5BgB2CCvU/88GTsehLFZEAAAAASUVORK5CYII=)
Adding (1) and (2), we get,
2X = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJAAAABACAYAAAD1eP6XAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAWkSURBVHja7Z2/b9RIFMfn0h50NERMqoOCjp3dngItFj+ui5DXSnVSCmvPOgGSi5ww6aI7cj13DZQUtHdI4R9I/gYoqZJ/Ycl5xuuN15ndeOz3xrPeh/QkIkWx5+vPzHsz4/ma7e/vMwqKukEiUBBAFI4AlP77YVGsXMOWtAWrPV3Rz0S72X+SZOemYOxr+gvfLwc/9aLk3iqJEHn8ub4t7Dsfjl9AX69L+k21m1TRbk6AHusd7STJzS4PueMhf4EFUNf102lHABFA9gFqo75Ydg+rClCbdVKd64IAVMj15/oQX7F74dgfPhacHau8zMXxg3Cvt2oA7YXbd6ZtyGoN3jvZDvfuWBlJLvSbmOgHAtDrQDxZVJyqEME7zMYnvniqitIgeqQKvsB7xBk7FX7ydFUASiLvXr/v/bF7cHAt+9m/nT1Q/M7XRD8QgALB3+QXL8bBwe41j/MPmLONON7ZlKNfuRGyYUr8ON50HSClk/B+z+EpQpU+yDMRvH7iqn6NAZK/4/t6QJQA3PtQFga6iEt7z1kZ0lz8KmC4WkTvhQ96aiRABKipfqizMKwHM9dzb7HP7Jb3+VLvzesyMXq/igDJQnYk2HvM9A+hHxpANtLXrJEaAZaJ4/osLAy3e+nQc8L63sfdOL7usn5oANlIX/kwq+slMwEqFKGuAJR1OnZUmICc897oLyyIIPRDAwg7fXURoOK9R8Hgmax/FETD8cu1AshG+roqTy8bnleliJ4+4FOsqTyEfigA2UhfXS+i81CFNBJAs5TpWhFtI31dtV5hsobiMkCqfRVG0bb0AwfIVvoqD/PFOkHOZNKGvUyF+VLl3l0ASLcHFce71+UIy73ouav6gQOkFr/E6NAGPHlk76rwU7l/oxrvD3yTBbi2AVLbFjJdqL2vULZhQ8IzEvyQi+CNZf02TPQDBygQ7B3myulCES5mLWoz1Y+S222nXKOtjP7W2+ILW5z3/9VtD1nQb2KiH70P5HgNtIraEUAEEAFEABFA1kK9z1RhvWjRbGmdAdJpRwABjVYEEAFEABFABBABRAARQKsIkO54klzil8v+VY7fEEBrDtCyo9Dl0K2+E0CUwiiFEUAEEAFEABFABBABRABNC+uqL3gRQARQoyCACCACyDZArvjbmFzfJYBc0Q4IoPt3b7Ab3+4nyd0qf0z626gjubkvEO+dDP3xYxsC5C4T875El00DdPGr+PGQ8Z8/wQNkpl9dj56m0cSXSKddLQFyJ4nSCq16kFV9eppEdjylni+RCwA19TiqfY8NfYlAAMoPqHEveFW8kek5b3SHMjX6bD38p+4Z8rYBgvA4qhMQvkQgAMkLDrzoF+0NqvPV1VJJ3chOb7Jzzvv/yZRpWj+0DRCExxFsSqvuSwSWwpY/XLwR6OL8eCFtpjncpNe2CRDU8WzIYtrElwgVoJk4yB6JMsIw/CnwBr/ljhYmQ3+bAEF5HEGAU8eX6AqAICzaxLHNdZCCx845xvYE9DQeyqIGSDNjXyK0UxnZUMgP2zilOu3VX6oKv+4AFa9n6kuEazCFaApQbVpfrXhvFSAgjyOcurKeOVdjgOTqpG5WZnUUUiKkxbTjIxCERw/ezLYFgGTdI0TwSpfSbAqglti5d1Sl57Z9MhXC4whlBK/p7lYLIAmIXE2VxVd5X0UtlQvvT4xhWPu9qmn9VXXtqX17l+YeR5D6mfgSgQBU8OQ5W/S9DKzFMAnKlnj4t9y7kfchGy+d8wdB9MyoXmv5bHwTj576ab65LxEIQLJyX35qAe/janO+QGls9b23Jt5ArgBUasvE1OOo9vS9oS8RmSs4BFBXtCOACCACiAAigAggAogAIoCkAHG82eXvxk8NE/AAyvTb6OJ343XazQmQbUquwXfjEfbtCvpNOv3d+JJ2nU1VFJZGKxKBggCiaC3+B/alcZIAdeEVAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
.
Now, X + Y = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANQAAABACAYAAABiKVlNAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAdISURBVHja7Z2/b9tGFMfPXptkypLUp6mpgWzhWXsGQ6Frp5vgUoSnAhoIV0jiABwMWM5moEn2IkvipU2Hjk1QZ+xQ+x9oijhbu9j+F6LUdyRliqIp/rrjUfoOD0gUxyK/fB/ee+9+PPLkyRMCg8HKMYgAgwEoGAxA1UcUQuYvsxrey1zC/cxBv3K1A0AR6/c3bjBCPp4L9mnc6InZ69+p0/30TPoo/l7IgLY2t6BfudoBohiHMIhxsNHv35jm+9xs0S1ZQE27fknaASIABf0AlD5AJeULKnKGaJxfR6DqpBuAkugQoXzhc7yxjzLf1NwpNq3WKqPkUMT2lB0uO9ssTbFBF6Bcd+PmuIb0VGauVUQ3ACXRIXZttnZJ0ioSV8Lsl1Kv1WL3hfPZvW/43x/a5gol5JRZ/ft1AYpfx5h2zNJWNwAl0SFsRp+a9sOV6Od7e90rJqWvZb5lgzd79MF6lanzkdF1b+oOlLiHxr0XXde9KquEX7ZuAEqSQ/Cfs6x4YPo98w6l5uvu3t4ViYnx4/O37FkUWvHdhJxNgkUHoDqMvOIhHqXsbWvdWVORO3kj4nhIGdLtMYDSrEolBDd7j2RdoxgBF8g7smC+i0I7zOtY55XOQPkOfDIMj3nuRI2jtut+WaFux5N0A1CKgVIR7g2hiXGMJKfRMYdyHOcr22w+8OEapA27qtINQCkGSkW4F4QncW/ToWNMqDDqNg/lvYjIgRipjM6+rroBKMVAyQ73phWokbBL0nQDgKoZUCrCvUl5UlJYoztQ3Lxq23ixpUTdjovoBqBUvmEVhHsj4VFiUSJ5DkxXoLxRxDiSMUKVoRuAUugQKsK90Tf5eAIfhDXM3l2rI1BispyaB7JeSkV1A1BKk2r54V7EAU7D8yZiSY2Yn2LHRfIAFfr5a+nmop+JyXKZFdKQbsH3Z9ENQCkCattZNiizn6q8Vj/fOPXXoc1vWk2Ll5/Zxu5qEadQoR8Hp8HuvWg77iK/dtftXuWfNe3eemW6pRidAJQioGxGXqZx5NKdw26uDydIKfvL6vVvFXUKFfqNXPe5NZbMn9Jee5W6aQdUwrbi6PCvZLt22TlAXQz7oeRop/xiQtuKB6El+5/Cb3e/4vIh9DMDmYkqgIJ+tQXKSwytW/6s+ICv34oTnyf53uJJehK3ohtAAaiqjA8KUoAKwrc8/ze81CRuHwpPUs3GktK4epaAEuXpFAtBoV827Yr94g22KkqNORczBiXMuHmBDqPP0lZd4BAASkugJp2PEGei5EjNX/kmsTwXx6tjYk9MaD5FlKDpyi8qciYABf2kABXZn5LdKDtsO9tf53kA3oJIerbsbBsi1MswQTrhMMLMlUIABaAqC/mi4V+DNN5nOewinOSJAoXR2ecz1lmW7yQcRjhmaX4vgAJQWgB1kVORwYLZ2c0SAo7shUmx2hchn2SnKDDqA6gSv+hiTRQZZC0oJO1VAVBqnaLIqA+gSoSJH8/Ew76maT/IOsrkBQo5FEI+bYHKU+ULjMNEKf0z70rhvEAhhwJQWgJVoMo38FY8sMMiIiLkA1AzH/J5R9ryeahic0bBCCeKEsz8rd3tXlO1GBZAQT9tgOJVPUqXfi9hZDrJE54BKAA1tUWJaTEABf0AFBwCQAGo6QGqqs2QCd8/XyegQlMfc1U99yw7JwBUJoe4e/s6uf7f3X7/dpqfd532okHJ0XCzJDWOWtbmqupuEln7K/3AvnhG6Ldvq9SPa+Q4baPVYjuUkn9k7M9Ka/4hLakO2EzSDhAVcAixKl5sPxmbRvicttdQCY4Q019p8tlyOgAllqo1Gn83GuS9v+OgEqD853gGoCoEKlh/SE17J5gyGNmJLLl74XB0ytlfSQegRkKoioASGnqdDAcAqkKgeLm/afa+jwVNnJEt5zjhsHUMsj/sr+SFmfMAKlvexDey8pdi2nPNAZQChxhzdHEWhtwRqmh/JQDltQblZylmaRQAoBQDNXw4kvvrBpa3v5LvFG8k6fev7kDxUG+JLv3MAcoJ1BsAlcohis2jeAlusXWNuUHO0F9Jp1OPVAMVhHpBSJ4VKJwcqwio4EFVcYJscO1p+yvNMlDRpg4ASkOg/M2VW1WtQQwsbX+lWQVK5J5G53l4ErzbbV8LgOI56KTiDoBSABR3iriqn/LrT9lfaVaBStg/F5xSfP5n9iFv90dAVNAh+MjE8ybG7J24f1N9/Wn7K81yyHdpIQkhX7VABdv+oyGEaM/itBcZM3+UdeDMZf2Vwsk2gAJQtQFquLlSLFkJr6O7MJkOwsEp0l9JF6BGNpca3+1XsUgWQGkAFO8vlHw0AD2RuVKiaH8lHYDSZXMpB2qFkj8m5U4ASmFRom6G7htytANEAAr6ASgABaAAFIACUAAKQAEo6AegCjtEsAQl61kNutllR1WLLicSgZoG/fJoB4hiHMJvmK28FC7Dko6qllGenib98mgHiGCwMkc1iACDASgYTEv7H7n4bbbzBpVZAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Question 20.Find X and Y, if
![](data:image/png;base64,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)
Answer:2X + 3Y
…(1)
3X + 2Y
…(2)
Now, multiply equation (1) by 2 and equation (2) by 3, we get,
4X + 6Y
…(3)
9X + 6Y
…(4)
Subtracting equation (4) from (3), we get,
(4X + 6Y) – (9X + 6Y)![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, 2X + 3Y ![](data:image/png;base64,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)
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)
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Question 21.Find X, if ![](data:image/png;base64,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)
Answer:2X + Y ![](data:image/png;base64,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)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Question 22.Find x and y, if ![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, on comparing elements of these two matrices, we get,
2 + y = 5
=> y = 3
And 2x + 2 = 8
=> x = 3
Therefore, x = 3 and y = 3.
Question 23.Solve the equation for x, y, z and t, if ![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
On comparing the elements of these two matrices, we get,
2x + 3 = 9
⇒ 2x = 6
⇒ x = 3
2y = 12
⇒ y = 6
2z -3 = 15
⇒ 2z = 18
⇒ z = 9
2t +6 = 18
⇒ 2t = 12
⇒ t = 6
Therefore, x = 3, y =6, z = 9 and t = 6.
Question 24.If
, find the values of x and y.
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
On comparing the corresponding elements of these two matrices, we get,
2x –y = 10 and 3x + y =5
Now, adding above two equations, we get
5x = 15
⇒ x = 3
Now, 3x + y = 5
⇒ y = 5 - 3x
⇒ y = 5 - 9 = -4
Therefore, x = 3 and y = -4.
Question 25.Given
, find the values of x, y, z and w.
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
On comparing the corresponding elements of these two matrices, we get,
3x = x + 4
⇒ 2x + 4
⇒ x =2
3y = 6 + x + y
⇒ 2y = 6 + x = 6 + 2 = 8
⇒ y = 4
3w = 2w + 3
⇒ w = 3
3z = -1 + z + w
⇒ 2z = -1 + w = -1 +3 = 2
⇒ z = 1
Therefore, x = 2, y = 4, z = 1 and w = 3.
Question 26.If
show that F(x) F(y) = F(x + y).
Answer:![](data:image/png;base64,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)
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GQvW0NseWc/iw4TbPJNud4Nee0QHHW2hBURNXFiLOGJ3tP3gTxZz95GZamAdOSHTYkWf6VLtN0nlFnA6VIz1lMBrJI28TajopuJuI8P7d2UeKcF7e5LoUVy105JzhJaEJuuG+t6BOvVRu9Jj67zGngP6fjdJ7xyq5ryqmmlZAVgvFvT+UzC5517KBcxFm3g9Od1hZpPRvP8vEdNC+yJ5ZpCSlWukvLRXnOHbtzR56lVR5mFuJsyO0PWXA7D7/BCwjxRwwsgttkAQmiW5oSJ1ZWNFKV+REbY9yzbUwIJx/4CX1iShh0xSvK6daG84ngVOeliszGq89pQ3CqE+kXnK0htc2+z9HfpJzwRjDe7B2dMTDz4KO1te5xx7b+SQc1zwbQSjafJej8UKRX8rKI3/ym8J+73uBMEWQLwdFZIk1GKHnmOZtvCBLbfSAGddCO4+XtYNA9GUS8ga8porHVZet7sZwWnwXpUfF7AMLLncHty6y4zWJlZJhTu7eIPGc+kbwLc7R5/nA0aBDRY5C5QH1k6xgbp1Rg7d7TuD7hCWFQ9CkdTlcYp3RCOAw59UJOO8WOV3NOtayELOGxSZTsixTkvtu+RgVbd2Nc9+AjCXovEARhfTzZPA92Ylk/bQXov6xA/1ZXXNWTCfnG1FOf53qZi7PjftnvX/hUOmTltcj95Pe7P2mluHej/NLnmeJAsZqYxW0vY27n5TcVtzm+/Tnr+iGH9CWUlvNQlcMb5C3TNLtxe7Yd9yvBVUyf2I7LBxacxlmKq2wjjnPa8TaE+KTl1GN8ekWO11H0VepC7A23fbVpWb8H7X9uu+vdPm20os/jPOdwWV2BE75YIntz5ZHuMpYfaJLrhlEFztYY1ZjbUZGbgeDUnNNW03moyyk/L2VUtfNSchPnMPJe0GuwsyBnU7AcWoOojFk2mv50HcUZ3NYPIacFveI8D6eeZpbGFKcVO2kw10oowYxV7K63iec19vXG3uzsWbuYAgJlL1MV+KPF7X4XwW0dCwiYctqg7VVCTrktdhByqnmP/PyNF1XkNFdxFrNxGZdKtCMyH1byZrWXvQVEVmUXZ3Bbz+iZbjSWnNNtE06ZwJGLz6t4PnfuNQSZl6XxFk8VMO/5EHUSZ3Bb0+iZpoCB06MhzmIQ0wNXqlymnN57v2P/Zdbr4EdJnGvFrdO+URS3VRDo2nBaYYuqEHEWg1gcSVlF0BIzhR6EUxFxBrf1FWhwekTEGaivOAMAAHGGOAMAAHEGIM4AAECcAYgzAAAQZ4gzAAAQZwDiDAAAxBniDAAAxBmAOAMAAHGGOAMAAHEGIM4AAECcgbqKc7QaOTgGl0ewvRoQZ4hz6Trneq9zmVcEZ+WSeF21BrgGl3VuK8/rLnc69h1CrDdyTU2IM8S5HM8iSj+5/Uv0//RQmqqURAL0uDzXu/MF2icirrQgRbP532bT+p9cRBniXHlxPn/2hHXit/PD4dkqPwer4WdZe9GCqUFli3JWUgHAZaYizSpCWQem4vxX+88PLPLFvyHOEOdcEJSMmi6Gu+gq10BqLkcxXNLq1ptoJ4gzxLkCEDX5rCXnZbT0E7VtaBRW1iKxQAKXkSopnMvd6BIcgDhDnEv7DFyAFeKcJNxABbmklbsVwg1AnCHOZXwGbl2oouNQnEtanR1QcjmK5ZKKM3xniDPEGeIMlEycwSXEGeJcwaWwYkAnLZOB0nKp9JXDn4FLiDPEuRqY8CJjNwTdp+C7YlyqNwSxuQtxhjhXCXE5sMLyqHLp+qOGGVyOwCXEGeJcpedQ5MCyV4A7xO+o9i48ygqJzIbzieBSCAa4hDhDnCsfcZERP4Ph2Hqv3fMH+T4irerBC7jcB5d6oG0UvPJuHVjL1773//2TrkBDnCHOxQi0275KBzE/LOd1rz88BZ5rweU2uIxfaTQt63fWThJIx9vQEWguzv+COJdOnHGeMwAc6UnQX6XQ4w0gzhBnAAAgzgDEGQAAiDPEGQAAiDMAcQYAAOIMcQYAAOIMQJwBAIA4Q5xfdAeDk3KlY1Q8BoB6IVrNXMDrkA2Ic0nF2basnWgCewCyHy0VBABANbHukJv+uP5DNdajL6ygwQAAAMoYZaMRAAAAIM4AAACABv4PthgpQ41LeNEAAAAASUVORK5CYII=)
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20fwuZmUTWTkWVp32o+j8uJac4oFJy6kHUF7GjvhRbO6zHZfJDmgFX6LnIuA62f/UH29YPrv6PHlgkR//ic/THaQPcOL8KfoVT+5KfEQEbplJH12rC71XP9DzSj4FNkZmGqPJsHqp+iEvTKfFWdib4cU/9rXygFwXNwwfB4NIwGxv2tR0SX+P3DLm3kCn9SKOcuJlBmiA+mtqzj1st9ihNOX3w/QkilKXNadsbvM93+tDR7LepRXiMaBaVftSDgawwis2dm/yYJGrmlFO2/i0iiCe1IzElp+yU/xfy5a3uzaSZtBgRl8MPI4FknKhl5O206ce0fhXZgPSiFh3BB/yUgVTtLZ3XgEO1c2x1d1yabph1eKZSiWJAFe/HYWpvzffB+dx8IGek40TtaBx/pl1Tu2MUtciDn+oBoFmVNr1ONdpO2BScpc1Z2qs/RGKjtvPbcZ9J094iRU1LpYk9e4y9WHa3XFPQmFbUtP49S9u/uT44kmtJQSNpkKOLtPJ11Je2u7VOwSHeD+25+2HEHkO/ZRmczlrU4v369cIsRU2lLMPP9OqjefWP4kgKfh4lrD0d6bPrLeHHV9KP35j8KNK2YrlmXj5IWjcL/i9N+nHakv7ElEbKh4HWFxhjf1LT5jxFbdr2amm222lSj2nbW/Sampq5BgvxNKgwkGkWopa1f4sQtaR2jM3ti9nLke7rUV+ujT6Ywg9/1NND8xa1FIvzx5Nsc5lV9aP066/Dfh0OGGYhamqGoD/T8+obHpA5P7ofx7VhyM/RwVHcjIbWzkb9OBQ2vbpwXj4IZpVJYm27X+Va/TgLkVCjgt5+74em/HjZRC0gDWv+mdJIab4/TXunX1PLJmqhFAwtmIuTJM6jabFpRS2mf+25p9sM60SqBDxurSYu0Bt8+Tzqn7L5YZR7o/4IUmETpEpnIWqxftXEahaiFgTTDM907ik4XrgR3pOl81OlFKMcMlWshvxIhSEGPw59sDRXH8TtR5P9d6LXARQrainXkKjCpsVaj9R7kgLOJxlz/ZlELeOaV5BGSijjz9reItOP1J623d7TR0Nxa03TiJrs399k7d9CZiZytErl+WoGTDMpZVN0wT0kTlqgT+PLJD+MW/QuMpDqtvKjitrsdnRGWpSoJfg1JLJalXM6UYspCadA2FDH580xNawXFUl+/r3i557vU2W7Ku5g7Q8/1meuqnLS9vZC/NSrIePW7aQPvpunD4YzxqFd1F9d3qf2YRrujD3QeHOz87pjW/9BxJEquTCLIC5Lpp+ZHySxvtbp9t/5JOOaxyRtzio66v1pUo/j2jsvUaPAzmzvUxXUTQGs3+9cEmIvRnVqpLjWtB6qFJ56TZSoy37zv1OVxE/SvwWleL4L9h2K/WKn+iBFCy5nYg3CWuC84pVp7oMxvtxRlZS89DrRD+25+SE6QpbVdHwLzJa77PKR8YQFLbMqFGH2+mfKr9HApvNTVfcRXFn5rAKjxs+zaLn7yO/J0ve5DjIEP18Z+SnFRt9LZrsfEj+/R/yk14ifygbFY2a7sX4c9UF3rj7Ycth2w2In8rzHha63vLZoWd/a7u771jTHZEUOB9aQbrPouJmS3MMUSqeNpuDMvzVMAYUfmmnanHU2SqfXZ565JuxTK1zUHO8Xvd61n2iHhj7X9+Wo6X447endifqX+m6k3/yHb7Vl/W7S/i0qDRnYTptbW8696L4ksceMSv2H7V12vF+OHK826ssD9V2iCKG8fgilIb3lW0GfazbMTdQS/Eojegp0YS66gxUxEBm+5gdoIdhcJCQ/zRuKqa/LcjAy34cX4mfbyE9R6q9s4/zcHs/PpYO4/Wdl8QEVtwT9m9DeidOPk+fp86n2C4o1Us6UZt1evjFxceV+2llXGU8UAYBcg3LFzn7kKeJF515ZqlLhgylnunmIWrCukMPp9PIU55kK5rj26tWJlDrKIqh5tBeiBkDUpoP+THfnXPVYHh90K++D3ERtuDN+9ik3nipMWaQxq/aq4giBbIvnebQXogZA1GaRnVHnco7f1FtH9GTBSZ18kJuoBcSeccVfHhd5pmkvOYqvN0TWnlKJZU7thagBELXpgh9ft5ly/bDKGAQ+yLZudWFFTc1+ZpmC5Km8nGY9VWsvRA2AqAFAgaLGiUxrVTM6cqWIg2Cr1l6IGgBRA4ACRU0JBZ2uPc0CJH2217b/0d7Yv5G7QFSsvRA1AKIGAAWKmhIKdffWJNj2nNUi909Urr0QNQCiBgDFiRoAUQMgahA1AKIGQNQAiBoAQNQgagAAUQMgahA1iBoAQNQAiBoAUQMAiBoAUQMgagBEDQAgagBEDYCoAQBEDaIGABA1AKIGQNQAAKIGQNSA2omafg8T4SL5ui62Z7FDe2+jCqIGftaKn40sn4OoQdQmItuW275hM+tAXLduH1zr7toX4YLFutie1g76u9vtNNtte48x60Xa29vjRI1zNGdRAz+57c+raruBc7eztF9xD6JWeVG7+u4b1ht/vToYvJv7b4mbvE8dr3ed/qZzLpllneZxy3np/FwT2+PsWHL3PhgZ9S4u/t/iovX/dJFkWlH7Z/vvPrPYB/+tBxbJ0b/4HP1xAbadRGw7IdvqLmx1sH0gOPe/GucyiZrPvU+j3INIQNRi0e9vvOkP+46jwU3cImwf13m9pC62T2KHuEDXOiu7qEnbjuJs6/T7l+rMz6a0Xfe7uN26eVQ12/mASnAOogZRy+93ttrMJxh7Gb2BgN9DZ1kv0wa9KsK3fYfPbuJtv10DO05NdlRF1NLYVtfZmrL9Wne3WQfbIWoQtdxFjd8Mftl6al12nkZv8hZFANZxdB2lLkhh+1EVbJ/UjiqIWsi2yCW/um11FDVuO7OeJNh+aDXXH0LUAIiaSbgMATEpWNbDv/WwPbUdkcBYBVELhMvQ/qSgXyN+1sp2iBpELX9Rk2k200g+CIi0JlPDvXJ1sX1SOyohajLNZpqNBYG9puu+dbQdogZRg6hB1DIFP4gaRA2iBlxsUUtYN0tKa9XDv/HrTVWyfVI7KpV+NAT2pPRcrfhpWDerqu0QNYhaMYUiKjcfWyjiPaijf+tie4qCAqMdlSkUSbat1oUiq8x6nGy7WynbIWoQtUJK+uP2Mqn0nO3t36yrj+ti+xg7Tk12VKWkfxLb6sXP0f1oge0VOyAAogZRK0TUTHuZ+NE8fP+afVTnA5VN+32qaPv+jvNeVjs0UbudMrDMRdSkbScG23Z82w7rfDjA0Pbux3WwHaIGUSvsmCwxIuSbPOlMuYUtd9mlh6nOo+C62d7NYAf9vzh+yTqzmmsP0xwyO89jsqRtJxeRn1sO226Q7Zv9Ftne9ZbXhO17lbKd2r4b5tz30gobRA2iNllw95Zv0cMiD0197vYGVy6KryO2H1TV9jR2qNE/f48GfzawU1ZRq1MfTSRsvu2LlvWtsH3poNPdfbtqM86g/RHOpRE2iBpEDQBmjnmLGnChuQdRg6gBQC6B5b8gakDR+Cefe40I9+CYSooabr4GygMqTohWSuLma6AI0Lp39KYCOAaiBgAQNQCiBkDUAACiBkDUAIgaAFEDAIgaEBU1Ok1A7iMKAT4C8gAFDhPfum122yRq4ChQAPd2IGo1ETV+EWBkb4cAO4neBAwAs4DY6Gzi3OheNnAUmGk2wGHbPndexXEPogYAAADUc1YHJwAAAAAQNQAAAAAoGf4GkpyXVNd7Is4AAAAASUVORK5CYII=)
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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, F(x)F(y) = F(x+y)
Question 27.Show that
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
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jvxS5eIVl0WrRo0dgnjeiEaNGiRXGIFi2asv0f5HVbGrWxppoAAAAASUVORK5CYII=)
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IAeYA8QH5NIM+ubOvyrmaVlngzMzSSaTMN8m3RJ0/boZ/l19FZgz5UgkZ7AHnFkMeG8UJ8nRsdlnH5QceZRNqzZslLS23TJ0/boZ+lW5AC4VehjkNnb919jK5qTLSbpRFAHgwMDGxNDEQAAwMDA8iDgYGBgTXR/g9z03N8BmkWGQAAAABJRU5ErkJggg==)
Question 28.Show that
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
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)
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)
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)
Question 29.Find A2 – 5A + 6I, if ![](data:image/png;base64,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)
Answer:A2 = A.A ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, A2 – 5A + 6I ![](data:image/png;base64,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)
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)
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)
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![](data:image/png;base64,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)
Question 30.If
, prove that A3 – 6A2 + 7A + 2I = 0
Answer:A2 = A.A ![](data:image/png;base64,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)
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)
![](data:image/png;base64,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)
Now, A3 = A2. A
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, A3 – 6A2 + 7A + 2I ![](data:image/png;base64,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)
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)
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0P8Qg4rl6h/kjJozj+mHpylFIXe8WhHbPG5qIKe0FuMM2tC4Gy6cLH/klBhjHUDomh46DV46QTsOsTMmJ2fcIfywp88t9l8kXVjI+JXz48azJ61xvDqlfAYSt3trTRJ2DhRxo98usM7wlkzYn51jh/mGgCDp9dk/tx9pDiBHGf4KeNz0ZWDiJgyAEEY5nlZhVQoixUvf5hvdk86waMKO/RhR1VCTSqX4AelDXuDiZkagk4S39UiTBd3IYSk03d5770QZz/b1b2Ev8fX4VHSQz5Vzl2DO2T6vKHJ5GAw9Bg5B7RuE8U1G0dJN+O4eU7sN1lTxLyJWBPPNyb4vHg2tRezXN9uB+5uc/nR3uQF8c1pIYScK/fURcFoyWgogi4H2QxEkjc2bsgOwkRhpFkmeRmmMw/EKUhOvMEmnYgYCqV2ifunnBRBAyr8Bqt/R58r2gCTuKPKktsQbpANzs08ogGHab/IW4jXRb2ilt+ThhD+zIC7q8EJX8rkoBFDlVIwH0yUbjnPl6+NiK/H8PuVknOBOyJh11S5fHQ6ZwbIdkAP7L7YOZHyDnYbyQEbIkSo7xJwg9qWDdHnefi26+SV/9xsdt9O08CtstnbFdVsnXLvbLVZtxkX5TcbGev9C5xGimSdEBnTcA9p2zlJvGiCdjxx8dlWP0G6eIkuIdOPD+3x3Lpl3HGMgt7xS0/J4yhYxlZepN83JXg1sXv3CDfufiX7AlY5NC/8xz6A1UE3PeJgla/jvwvvPKlMZwjATvyPtW88XBhEA/unrC9yu1pUgKOoGuX+41m+83UCAFnNnvLcQXcc/b9wvYfZSv7VZ3eovrqLcnqXsuKCIuSqxI90bIPSZP+HzgkrIbDSDitDcb5hXPkqO2dEEaxjUwvS6frcfUKmgxE3Q9XPVbjdAFbrTVmlyqEHNr2le9jqbLXuPJzWrljCdj2k9il2LAVcJpKouoSdN8nYuz1K/FJLt/VVxw54os/iGFvybeIEjTIbZmzP4HOZuj0o3MQDz3N3ecNIuB5W9exR6ZO1B6wLPnKIFvZa2KWoz33/16WrML2W8PKxEXJVQVxBg7I01mxDHWgi2NI22eytv04vxCTAT4B8Ca1ZrN+Tp8mn4suy1ZrWnaYHzoXo+vVvRzVPlH8Ma+xiqNLf38/YNISy14Rk3uU8rMj95ukcsP2ecPK02kImNt9ROfASuLcgHRUE7CM6PL2SZl8MUmyY/i0N4ZtUqseLLZaNM8LOdx4ABJ2JwBh+7xh5elYBDxJe8Bu84+s8zPrMtbQXlFgM1TjXpnlZjLBEmcpSc97VAV0dPeFVXWCh/mFjBgGZw21nv1z9WnW2wMyfwQdk/pj1okuiS6iMUZRI2NW5eckMWTvz4U0YYmEeLKbsLw+URT5BskPmrD4Yngv75K5yGuavaoN2+d1CXiKmWP9IZSA0+8B50PAxCmDMmbelvxtSkXyCDp36xzfeKbqCE5WcrOAnaBHj7N4Gn3u503AgROXHPeAB/7Y2Ezqj1kmuqS6iCNv1Hiksps9Tvk5K7lB530HR6+2FhPlqQkhYK9PSG52msrTJ6PKL6oE7fczDeLBmbgFnfe1tzm0I2YOjiwlIuAsnCdDAl4PGky4GcpfdgS0m9feYsy4k3U3srdE53V0z5WHe6pWDYMzZunkZoGWSGTkhf/2IPuGLC1y129KAh67ys6TgMP8Ea7gi+qPWSW6MF1qNePnkEzqkjtsxRiKizysmyrt5R2j+Zjdz6ljaG6wvy1iaM6JoQQrq0khYLdS6C/x+n1CpU/C7VFx5edJwCHxsOMtl7tlaTq3eos8sMvQzpWZbb5SfhplzEp9FzQMin3tIunZtfbhUrcIGpMt29cMyu+TjTqjTpI87AqBOIgN9zCfWjNnTXFbSoxVaFFys0sOZE8j9Ni5e1vs1TQM2lU1CYniF0URsNgTH+ePGZ0rzkoX2LcXZVdxjKxZc8bwLHS2w/lLlUQ4svpkjbuJV84x5UKZU4NLFMT9z+SUxWMIGs+SNiOV4S5oeI+f2XHxgly4/jub4IbzpbUkcsVxsE8M3+qUtU8mlZ8XAYtrJ2H/1o6HWTce4NY2iAdZuVzTKof2/c99P/rae2HHRBKwmF3YnZiBpe5WY3YpvDReOTTa8tt2ZKSWpFmmNegYhQvdH8c5/1qk3MzGmCdwvVZ532v3Sk1/P44+cWwQxS/kqys4FhBt77cfXAa9EWsFGMUfc7oJK6ou4uo9//hV2CeyKyKztpcXsL8WtbktK7lrQzFUTRVDRROwExdfhzV1QTdveDNY5VDlTVjj5U8HynePBYkYVrj3K3IFq/zKq6cdD61LQWMruqRd23M/qlsbb0T2G/w+YAQCgUgH/D5gRCK/QQJGIBAIJGAEEjACgUAgASMBIwEjASMQCAQSMAIJGIFAIJCAEUjASMAIBAKBBIxAAkYgEAgkYAQSMBIwAoFAAkYCRiABIxAIBBIwAgkYgUAgkICRgBFIwAgEAoEEjEACRiAQCCRgJGAkYCRgBAKBQAJGIAEjEAgEEjACCRgJGIFAIJCAEUjACAQCgQSMQAJGAkYgEEjASMCIySNg/t8UxykHU1E/1+OOT0j9lEpjESFD4JTXYEXr5dPPtd9U1slgEnR52eTKUB/Ejxda0ucyttNQfIfJSxJveeo3CQQcln8eDL8rx4Oponx2QmLIwag9k9qyNAQMxt6w6m8wSh7zf/9TgFa/rFudN6N8dk2n64SwA1WrbZBhWXWm6+w2peQrqq+1y6DXkH5L+mLffpTtzlkbrAgnLkqXl02uDJ3OymtVQg647N4A9NhodWeSPJeljTpW/c0ZSr7sy6MzT/Ql64qMXJPEW5765UnAIocmIGBhwzmRf/b9n+8062dNRt+rEHLo5ltavf6LerNzrijyXTX1y4xqTgzN7M41O7W8Jl6jNif7w7HhBc/nHns2bVveHbbl0i+j2NL2G2UEfPHtV8mr/7jY7b497tkNa65aqxk/b25vvyI+2zLP2wmtejDO+fhn+aPkmUqiEzOVSuWvlQr5GwxClISQh14uNs2ZqyKBNm4swM83GsYCl308Y25ezduBHV2OfLocqdblZZMrA5/wtfsTWBds+YOkz2UFiG9KtCOvPL7ceQGxBHbyE1d3NN7WVdpN6KdF18+P/2Tfukvo1U+zJ2CRQ//Oc+gPktmcHPsJo9Npnpmn5BGh8zt1a+MNIae98j2DkofatP45J46zuZPeErtCtMqh0WhdEgsXHkMVjRzO1Dd+lPdEFkhR08g3I/Hh+gQz+xMt15ZaQlv+B/cbTfjNYMWcOwFvbzdfMZjx3y759j/fMmaAwFhjazFsxs9nTbt2sMQjOrs8E2+GJWYsEQg4jV5xdXNXM9Ro3fT+vmXQm1EmMCpWYGl1iTs2ky43G9vXX2eVH/6m3myeDUtaUZ/Lyk4Q3/NU26GGebs/wXaTlBsfAXZy4u1FXALOS7+yErCTfx7L9LdtSo/nrI3qkK3bxgUgbGZ2rxRQtdmnutX22m/NoDc0Ut2vd7qv56nPMqN3jIZ1yT+Wtp/Qj71289iSSWx5BLYM84lSEPCYWfMxW9m6HFS24MZ6jxqN28Y0+Swu0cEqIG5pKwoBp9Urrm52mXu0fOhOYFSvHlToEndsSir3OC/b2z5H7gvfpNVP9SVrUbZXFfW5LO0EyWjWWPupdOINKzFCnwWVvpMScNb6+cmqzAQMOkBJFCYU/fzjIWCLTw41CWnYpVdtjzU2F/Ov2gROCI5g7PNaBYMNTHPrgvRvENN04aNmp3NmeJLAdRfl8oGOfVuaG5NHwO5mvMmm7oeVxbomu0JZ464IlBIRcFq94ujWlzFtfDZSQRANHOQA9qbycN4IuuxH1SXO2JwEuanLZjxZefagnrt7mPVWazrJcypiR77agMkA2w+KjzwIeKx+2ugeapkJ2M4/y3egPGpPIHwrYCA2jRyRC9d/B/uUbrMbfA7K0l6CySVngI4QQz65Tgztkeryh0XsBUt9yrBuesdYxJPG48m25bddW24IWxoPx9mydAQM3WTQfMGXvk8I0/8QVEOHskWN1j6CwC0TAWehVxzd+iQrIYEwglA1e8xKlzhjcxLkZoVWq/WvjUuz/wUka5cfqweyEl7U51QSnCjpgY2YeS+IuIokYLvkyMkhRL+yEbCTf34PhDtYwY+W0Jer5LeaxidgfPJ1rdn+PpBvpVL7o9nqns+T1PoTVQkBe8kZtkuKJF9Z+TnIluCzlQr7o7snPDEE7Dj8jnd2Dp15zc720MD0nBKv0bZLBWUh4Kz0ikXATolVttJKIz/ReGeoSywiPAFy1SQMsiPIVawg5MQQ9TkVBGdvMbHHYfYpkoCj6FcmAu45pWe3nB9GwPbqnt5xG83oBdG1m3/zlbNVI/M9r/71Tuf1vHUb0dNXfh6xpeZw1oVoHdClXAG7hm81Zpeg/u/vgHSO9ohSQJxk1+ufJxxgVae3qL56K85ZyCACTqpXWt1OAgGnHZss5Vo6XY8p97iMBDxUwhvTRDTuubR2CiMLttK9HGHCG0rARepXFgIWR458JdIwAu6JY598gkHpn90qSBziyAruPu8YAt4v+iISWfk5C1uWugnL7crzJjHxonxV7A2qZrN+Tp8mn4syWqs1LWsqsURX6khLOZ+xaM/9vw9q+goj4KR6pdUtbJ83rESqtJwUU5e0Y5NULjRQRJfbzUxunrC7scef7w17LpadIjTwCLLg8ryT1TQE7Oj3TRH6lYWAxWTQzj+nvflHlPh5/llstaibf9zzwXxF9zGs6Pod37CCy/kY0vA+r29xEVKeLkv52WPLjwJseW5iCRjQYOSel4BlyXpwZk/r2T9Xn2ZdbhxHwFnqFUe3oSaGwCasxr28HDUrXWI3Q024XLVVCbY7zu+iPpe2xAsJC2JI1nVchhJ0XP3KQsD8fUcmIZL8swcrSdFsxX/vPW4EvrxcI7/VHFvn1XUMchdg2zG4CWs/yh58UeVnxz+fe48bCVs6+8JUt25N7DEkx7HWCTUeha0iytYFnUUZMo5uQWdOnWNcz8JW9GpWXCG6hJzpTjM2JZV7HFWusuQhujEXdsatwqM+lyZ2iFOqY3wZKvnbVNEE7NXPnzTHXYtbxos4gkrQfFHzgXO0ikmfV3CjV+hYBZz3tSeF2tG4ozxFlp+XuS01Mi21pZhYiEs7euUn4Lrk/k+4Ls2g2qNxpaCCCHg9UgDkQMCy83KeazD38tyDHOgymPkl0SXu2Ey63LSw76Idvn8WftfwNOXEeS5rO4FN4MYw/zYNAK6A9N6CVwQB+/Q7HVW/SSRgm/DIi5kluGVqQA72rU7ao7xXnO5RHjoHe/a9fgxBaVfT2F6R+7+2DelHQbEBlQfY7hA3dj0YxJVry6kxtiwFAYtrJ6HUQKtP6pbFnP2Ls5AU4DybKqKLE6TupfD2tYOkR6rXPxx3SXxeBAywS+D0yLmD+NSaOWuKm1hyXP1mqUuSyZFzKL4v1+JyoSkiaiNNarn2YfxEctMCYqXCfviba832W278QGfmbKO1lOS5rGNn1WTL9rWs8vt1ZU2NoN/P7Hh74cTb6ajnQbPXb/iWpkkmYOeu46dwgYR917W9X2watMuffVoE4Qki0yqHF5vt2aEYWtpcLHL1KypZ7N/uBB6T22p+lyebr8RlHD9eBV897djytqZVx9qyNFdR6rXK+969i0ql9ol7t24UR7OPL0XfY3WJImqjhe8Cgz7CPp9Ur7i69cuhjdmlvo6UPc77TF+wLtXYuiR5fxHIKeUCmRYhN3UC88qH+Knp78t0iPpclnYCmbL9yQEqh/4VRpJ4y1O/CSLghyL/+IgAmoXma5Vfed+7UtV/XWjOMGd/wsf8azeGopyjVQ3Yy4XGvbBxDbLlxJ0DRiAQiElEGQkYUX4gASMQCEQ2BPwnJGBEHKyxb93R6JU/KSLg6N8HjEAgEJMK6IaHPWcV3wfMCNsp+iIKRH5+gwSMQCAQSMAIJGAEAoFAAkY7IwEjASMQCAQSMGJSCNi+B3nkUvRTRZ7vQiAQiLiQffGDOLcK9wErIuAqmXlk3+WMOfRl8JuMnYc8DTxb53xtHwKBQEwCgu5cFmeU5+TXFaZfAZOvgnKo7AsCEBPnN2oIGIFAIBAIRHSgERAIBAKBQAJGIBAIBOLlwP8Dud2eVUaNW8MAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, A3 – 6A2 + 7A + 2I = 0
Question 31.If
, find k so that A2 = kA – 2I
Answer:A2 = A.A ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVEAAABDCAYAAAAlHzZmAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAABD/SURBVHja7V29bxtHFl8q6S5xmjQxNKqSOEhlcaQ+hUGvENuAC8IhCVUGCIPgEbBpg4Vj0+mUs5Jr0zlOEQUp0gV3ke3esf4DhyqviJn/wOfj7ewHtVzOzM7Hm9ldagI8ILal3d97v7e/mXnz5X311VeeM2fOnDlTMxcEZ86cOXMi6syZM2crIKJNz3vLY1vNBdyZM2dVsFmgV2wtm9WMiOh4vPsB9rzj4CX/XbaNP/3hg/Ngyh85WLP1e0VirlJ8iozvqnLicq8YjP0Guhn8zOuslgW/+AY1+rfTvw8qonWv/mR3PP7AdADu9i7U/c7NHdnfvdnxdy707tZtJ7MOZhVT8RMCY1HxXVVOoPwcD/xN+37OamXF2FfAmNjAR7cCER1WVkTDYLfw5W1/cF31GW2MHuLW+LLNVk8Xs2k/ITHaju+qclJ1P7elc69fWowrJaJBzwEjtPNTd2/vHdVn7O1139lB6KegJcI2yILAbNpPSIy247uqnKyAnweCuVcvO8ZSi2jw31qqYLvG+9nRqPvuxfXaU7z74HPd5z3YxZ/X1i8+7Y5G75okSgSzYg9jwU9WayripwrGqOjefKvo+JrKI1N8FJJ7nQeXis29PX7uodoT3Ll3qUgumt3RmcqJaFSPan6Mkfd8XrRF9RfN3ugc8/lBl99b95/SWizZ58WTYBMytDKZyDzMqsOzUa95ro68o+D/Z6Gh+lHjWu8SLVlE/JTFSN7Tb6DbnoePWVzbii+d903lPFLlYxN5L0742Iz5oNfbkthstu5dMZ57yH8CnXuivor4CYmRg+9yDhd/yHBRGhElXfitLf8fSfDGg9ZH0YdQP94djc7Sut7+uvfMw53vIZ5HrIO974OP6ZmpYUSM+SkLs/LwzKtNM7OF/yMJw0qEuZ+Uno8KxgiD9xdPRPPeCxiPgPedryXzCIyTmI9XMnzYiI2h3KvzfKU14oGfj8IGKyf3IGrEIb6aHL4TjOIjg1KIaBg87H+ZFS8yO0c+TtrwIyKQ/m8qzwu78h0ctKDoLzJLJzeMFZvR42FW/jCCoQ/yW/fTonEReYdRstBFjeenLMbRaPcsRrXfee/Tia90PLb9u1B5BMLHcPfDBT4YDXgVcy8Ydh9mffWR9xvPVzJcDvyccnJvCsWFCr4MRlwZEc1p6aa0WlU0fER/+YPxJsTz0h8bWfMl+swAx5AEUPRnZTHz7MHQP0+bwZy36Ix38fyUwUiSpo3RN8jv3I/exxdRlfgC9phYeQTGCZcP5D3hvWelco/ja+znlJN7U1tcsBqsNEaRHnEpRTRe+LrWwmuPPdx+RPuZNvYe5320Ms9LJ3LwM49NJLIMZl2L3zWhvYvnpwxGUr9CuLN/ItpiIioTX92aGGQemeIj/eGektybevX2D1mBmmNk9BBB8dUCfIz38DBWQkR/DpK+12vicKIEN36hzZKJfrSiz0vjDXdYSUwyiCayDGa4+hddOFh+ymAkw/gttPUj+TlhEVWIr6qleH8BkUcm+TiFuTfJ1kUXMBoU0WiYH/REcYtZd01hFKpRl0pEYwcP4yLwG1IARvUvvs0uiRBNONHnLT1XItlEE9mmgETDV/w8Z6Z8yU9RjLN4GJ8MuyRFdGJDtJZ5v/ZP1TwyzcdSbARFpMK5t+TnAkbDE4/QXJRyOE8+gkFn+1rQpX4VfgCZ+sl8WCiYFHnPE60l0g4guNFAd1Djxp28w1VEMPMPOOAfeJAVON6aR5afIhjj5UxD5PduyfZ08uJrQkx180iXE/L7LYz289agquRer4Fuly33Il/H/Nyj1CTnQ2iOiMJxMc7nIqduWnoRTReHw8Bme0ySIpr3PNFE7gXBoh0+EMTuTfbvsx+NCGba81mWFjGewEGLaNiSB735dMJ2u833GmS5mVc/bg4G66xF97ZFFCKP+n546IQSJ6J8iMSGhoOZe5mF6qb9THwNxUQk9xRFNMb4WpWLngC+lRNRYuH6OSARZT1P5yMXHlJpYBYxmf3GOiLKEvpo2UhtFv25/hIqvlAGnUcifJBlS6L7vy3k3tTUOlQZX1mz9yYxKnGxSiIaLmXK7GBQqV3ynqdTgJeuSxmoByanD+GgC0L5tzVRP1Uxyg/n7cwSC+YReI1Wlg9LuTcpXe4t10Qn0BNLaXzZMocAF5NK1USblH2to27zTLR4ebELLpJwMs/TKcAXPUNK/LvX2rySHWITG3avfoKx/3Bp8TnA7LyKbzYmOGi878X7sal5FPY44D5cHh9kCyKNDyu5B+yniK/pHYNLYknLPbNcvC2CbwljVWbnw615BDSqHzV7PRzX2c50MNpHuP2QPTxjLeiVf95CIktsjZNZq8fDrJok/RZuR1su473BGUP+4JaMnyoYpUXU0DrRE943j652u1u6eaTCx408Phi5ohKbonNPxde0n9meYYIRYkebGL7ekHe+BA1jaUWUfISNrY3v0jW2jY2tX3kHsg7C2hw9KVSel66R0YSHZaLF6jzMKkZmnvkF9o0/uTuWKH6qYDxZUkSvherEV8Z08gjiw+0r8qEaGzLRIpt7UFtu+3q5N6UJrAEuXmtwMWWJbOVqojkf5BTyVKDwBBmgbWe2MEP7aRKj6fiqWDJzX2ZOToufCUbTp1lBc1FZEZ3XUOrtH6CeGW4HAzwqzAZmaD9NYjQdX1VOdkgvWnCLXxGcnBY/S4dR7hSnat6xFB6gyjnaTsZMHZZsErMpP01gtBVfDU4mZeZkZfzMOXB5XAGMS3ElJ3Bl6qeVEdGk5YIYokSHwe4cmu4lQWI25acJjLbiu6qcuNyzifHiocxa1UqL6LyOQu5j0VigS1ofcs8N5BXOpjGb9hMS4/y9JaqFMvw9kLkWApQTS7Ep0k8foQMRPyOMfqkxrpSIJkHHuHNf9W7rgb993fYwUweziqn4CYGxqPiuKifV9nOskHszaxj7khhXSkSToKveO2+rBwqFWcVU/dTFWGR8V5WTSvqp0NO2ibGviDFXRHVPTbEpos6cOXNWSIPEE1HdU4WciDpz5uxUi2iVhvPOnDlz5kTUmTNnzlZJRF1N1JkzZ840RNTVRJ05c+bMDeedOXPmzImoiMX3jNds/V6RmKsSnyJjWwW8Lveqj3FlRDS5BkB1wT05w9B2MutgVjEVP3UxFhXbKvBRdO6RYw/t596sVlaMfQWMKyOiJ5e0Da6rPqON0UNy6IHNVk8Xs2k/oTDajm2V+DhtubctnXv90mKshIhGqwPo1++mLeg94PCwDPZd6WuplQTMy6nIQRBBS4RtkJWHWeHDWPCRlQQyfspiZPFlO7YqeQTNhylOViD3DgRzr152jKUX0fjO7tt59/bwzmKMhmfNjzHyns9XEqD6i2ZvdI4aDHKW4frFp6ZPV4I8P5L4SC7dqiPvaH5/DKofNa71LrESRcRPWYx5fNmKrUoeQZ/nmXCyibwXJ5xsxpzMaqXIvc6DS8X6ucfPPckzPRUwXs7DKHqKVGlFNGwtw0umci4/I+f/MW5IJK1Z+ka/8CKzUFDpBw4nF1SZPsuQh1ktTrXp8h3w3ox1zYKIn7IY8/iyFVuVPILkI8XJKxVOTF+NEZ2XCXOCfthb5PjJuwSO5yc4xpoSxj9EuTAsop99+r73/n8+G48/lWstd89iVPs9cpad/PEtk89ot1aG/4b9L5eubI0vBGO1xOGNg+v+MxkCo+GiWDF6fjOmxI2i3GeFVwG37qcbCnJpXF7s5n5Sej6yGEX54r3TTK8rHxckH0xOhrsfLnDCOLVdNj4yeWci94Le4mHWTx95vwn4+Yh1HXEao26NWB+j2Mjg7/hv+x668u/SiCgB0sboG+R37uddwxu1hGxB5LSeU9bQLWxVJG8clLm6VgUzswUc+udphfeTK4zZt3by/JTBKMOXSmx1hnERrhYXFyQfuZyE96qrcaKbd1ZzL8/PYLhMLoLj5N7UFkZWrFMYceVElHTlEe7si9xlHtW6xK73jdeArbXw2uOg6/2I+f7k6tpG/7YJESU/C3ltMsvCy7Y8PGEOYTl+ymCU4UsltjrDVhFctvjQ5QRCREuWe1NO7k2tYKxh5l1OaYx5PeJYRP9VipooGX5toa0D8vMiH2VM1nHe838OxLPXa+Jw8gU3fuEVjJNEJjUOEyIqill7SBkOifIbC5qfohhjvn4U5UsltqrDeFFcNviQ5GQqGh9ZES1Z7k1pt3rOMRq8qC4a5gc9Udxilgx4GLNWmiuTZ/HwKznpOy/5RT7aVMAO46LyG1JQRvUvvmXNDsZF5WOZiQbRZBbFrGvRkAg/507IMfwUxTjnK+4xCImoQmxljeBqYbQvgssWH7qcQIhoCXNvkq2LLmA0KKKSGHNr1KUQ0XgZyjB9iEke6bIJR5436GxfC1qXV6GQMoZM8+cy3ks72epGA91BjRt3KKdb1WQx656clYhb3nIdlp8iGFX4WkhMQx+yLC4bfKSFXZCTCU1EaDh6DXRbJO9s5l7k51jJzwWMFOGC5WKszEUpRTRsGYLeYToI3W7zvQaZeffqx83BYD27WHo+NJTs1YQ3CZJuem4Pl147op1sVQt7uLU32b/PfjQimHVOzqKJSH7PZNFPEYwqfInEFqSHIYFLxNe+j27qnGQGwQkLBzPvKOsrTfsaXUYXiImon5SJnfkQmiGiMb7XOlz0NDGWVkRZwhEtQ6jNoj/XXy70mBRFlFi4lERRRHWGVTqYRXphMtvkdERUhS8bIsoSARYuk3wknJAZd11OIIbzpnNP2k8FES0DxlLXRGVrODpJEc7qMxbzVlFEk4MzcNAFofzbGrSIqtbcTIuoLC7TwgLFCaCITqHX6Sr7SVkGZQNjttQhgnF1RTSndkmsSdknO+o2z0SLoeldepUCvLCICmBWSZB7rc0r2WEssWH36icY+w/5jQWjJiqJUU5EzU/kSNREwWu0PE7IFkRZTkBE1ICveX6mdwsy/VyuiU4gJ5YyGN9WwDipRE1U5YPL76m2PgoJQfWjZq+H4/rYmQ5G+wi3H+YKXQVm58N6Wwu3o22N8b7gjCF/cEvGT1WMZZmdl8F10tuA/Whv5HHCyBXjs/OAvgL4OaHmnnWMvSFve6pIaaHUIhotTVqurS3WNulDH/L7ja2N79K1sY2NrV/zziacJ7LE1jjRYnUeZlkjqw34BfaNP1nv4fmpglGEr5N3ml0nKoMLko9Q2EA4EVwn6qObonkH7SuUn1kBSzBC7GqLMb7WwDihYayMiAqJSDjBAVtfm+8aYfTgyogZ2k9TGE3HVocPG1tRy5J7Rfua2g00LD9Gek91ZUQ0cRTyZKDwBBmD285MYIb20xRG07FVsWTJW9F82IhPWXzl+ZlgNH2aFSQXlRbReQ2l3v4B6pnhtjOgY7hsYYb20xRG07FV5WOHDPcFtvcVyckq+ZqXe6XCKLBKoNIiGrZc5ABVxvmgsgZ9OK8NzKb8hMZoK7YafEyK4qOA3JsUmns5By6PK4BxpUQ0abkghijRYbA7h6Z7SpCYTfkJjdFWbKvIh8u9MmK8eCi6VrXyIjqvo5D7WDQW6JLWh9xzkxyAYrz3A4DZtJ9QGOfvLFEtlOHrgeiVEEY4sRSfonwlfvoIHYj4GWH0S41xpUQ0CTrGnfuqd1sP/O3rtoeaOphVTMVPXYxFxbYKfJzO3Bsr5N7MGsa+JEZrIhod/EA9baUGnRiq987b6oFCYVYxVT91MBYZ27LzcepyT6GnbRNjPwcj6xSp3gVympZREfVeMhe6VujjcubM2em2foN9ihS60LtlRESdOXPm7DSaC4IzZ86cORF15syZs2Ls/1vbJsXoXjybAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Now, A2 = kA – 2I
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
Comparing the corresponding elements, we get,
3k -2 = 1
⇒ 3k = 3
⇒ k = 1
Therefore, the value of k is 1.
Question 32.If
and I is the identity matrix of order 2, show that ![](data:image/png;base64,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)
Answer:We know that I ![](data:image/png;base64,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)
Now, LHS = I + A ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
And on RHS = (I – A)![](data:image/png;base64,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)
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)
As we know,
cos 2θ = 1 - 2 sin2 θ
cos 2θ = 2 cos2 θ - 1
and sin2θ = 2 sinθ cosθ
![](data:image/png;base64,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)
As tanθ = sinθ/cosθ
![](data:image/png;base64,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)
=![](data:image/png;base64,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)
= LHS
Hence Proved.
Question 33.A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:
Rs 1800
Answer:(a) Let Rs x be invested in the first bond.
Then, the sum of money invested in the second bond will be Rs. (30000 – x).
It is given that the first bond pays 5% interest per year, and the second bond pays 7% interest per year.
Thus, in order to obtain an annual total interest of Rs. 1800, we get:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARAAAABzCAYAAABQHV1VAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAA7uSURBVHja7Z3Pb9tGFscf4yCXbZo9tMA29aiHbbrpnizSvuegynRtp8hBdWRBvRQgDMHLhasEPBgp1ZsXTXePW6BAm6SHdH3Y0wLbJM5lD9ugPRdFE7mnLuDE+Q+axssZkjJNkdJQImX9+B6+BwcvEsUZfjjvzXdm6KOPPiIIgqBehJsAQRAAAkEQAAKNuUpEU0R0MkbKMF3rNtGJ+GvdPoH2BECgAcq2q69opDSdB/CXduX29HpjZpiud73INqKuVSH6lRXX62hTAAQaMEBUUu9VbfuVUf4dDljqAAgAAo0pQGLSpNTSIwAEAIHGFCANq3pWJdp1gHFwKLavm3YeAAFAIACk68PdVrdQV79ACgOAQABI58+3Sq/mc299VjKsM87AQ8nqdwAgAAg0ZgDhNY7LebrFUxbG1K+LK7VlotIUAAKAQABIVzXq+kyOaM8ByDM+zSpqHyz/Xcm0pgEQAAQCQKRkmubvK8W5P3OYOCB5TpT/qWTZrwIgAAgEgEhra8t4QWd0R0Akv3oL07gACASAJP4+TaFHRNpu1bLOAiAACASAJEtpdPYBfCAACASA9Padpp5npH6b5ncCIAAINCkAKWvLxObvGltbLwAgAAgEgETKXXq/fSJYLOX/VtbY9bRX+QIgAAg0ZgCpOKDIaW99dsmov8kXzxmG8eKqxj6eq6yvpP1dAAgAAo0ZQMzy3LuekUysf8mpxU/Lpn0ui98BgAAg0JgBZJACQAAQCAABQAAQCAABQAAQCAAZIXFzmgOQK2hTAAQCQBKrUdGWSFu9iTYFQCAABAABQCAABAABQCAABAABQCCoX4B0OL1uIAJAABBotAHyy3EKAAFAIKQwSGEAEAgAAUAAEAgAAUAAEAgCQAAQAAQCQAAQAAQCQAAQAAQCQKI7KJGyvc23K6QpvmVht9iSiONejtJU3FkwsnEACAACjTBADpwH2zAuzRaL2oeM0Y+dltDz2LWV4pLGlAfCy8G0BwXDmg0fsi0bB4AAINCIA8SuaouUy32fy9EP4tDsDgC5Vs5fJIU90SvrC/zv9Yq+wBR6ki9fuxgcYcjGASAACDQmNRDxwHYASMOqnlWJdplufhD8d/dQKXXXPxtXNg4AAUCgCQLIepFdiTp9zj1Uivb9zX9k4wAQAASaEICIQ7SnaYem9Z3w4VHibFyiJqmrtyzLOC0TB4D0AZDDynSkFNwwKEruoU5x/eZw5iQLgHgP/24UGAJwuV/bLL0hEydzgh0AEgEQtyGUZvTqxNxe2qd7pfYDHLAlqaBDvd7j+BeIkxpsRPUb5z/9GnzoMwGIm348jXqgW2Agbbdi6m/LxMlcGwASA5BRM/nwKbnNWkHVKxsLaMzsxGcqlmqbalJQh3cvzxAg+13A0JQESBMAmRCA8DeiXZ5bntPX30dDZi9xTGSCqc6BAaRD/SKY3ph2+ZxMHFKYCQGIM/LQGFv4Ks1T16F48bf0AmO3C7VNbZgAIkYPjO5F1S9aYNAqN2TjpK8HABldgGxZxun5aWVHqzYW40YnRwvCne3KBwmszQcZ2KX7tVYPSo2qtqhMz+8Y1tbpYQEIV5yPg6e3jJR9v5/Ixg0DQJJY+NuL2PHxadv9RxIgwk0YM9zkP9Sqlf7g9IhvvGLeM4Xlv71k1M/HAaG2Ulz2rc1KB2tzODYNu3Qa1upBSbypFWpy1+YxASTSp9Go6zPcTcoKtat+RxdtVWDO39oj//tk444bIEks/JZROlPWXvskR/TY7++vqZf/WjKsM732tSR9MgyQnWEHiDeffz9uuMnfJrOz83/x4WLXq69rjB7wN0/Vss5GwkhYm91C7EbA2twtdj0mtpNdOv4zu8cOgyoa3RBpgGWdlgDIleBDnxQg3uzPSWsl/w4HCM1c/jLuDctnghTnProdnU6ulbVV/lBpVXuxl7jjBIishd8SI3HaITZ/t2za5/z+7qRqd5Tp4v3wSDG2r61sviNl9w/FhQBy4Y8v0Uv/u2Dbfxzy2ofKq+lapbEUlQ8X54qbbTmuN9UX/j/drM1Vq3FWNrYXu3Ra1uqBpjFiNMCeFmqbarfY6vlTN0+dr97sFSB8xOA83Hvh6WHngdqITGXKc++24pn6jf9Q9Rp33DWQrhZ+ry3inLXB/n7Y12pH+lrN6WtKZP/tHNczQNrMZtvu2yD075kNvz07cttN60h0XnBVeI7b9jZyrc0hj4sPnODbUza2i106+jMlYtNSu1kwWfslubZ+ATKsGhaAuA81h7mtHR2ZcAgozWB/79fuL9K/CLt/IoC4HYAeKgo9F0NKUp4zfX3Dq2I/4n8T5R5naTpb1eimrOHHLwSVtRM3nQb/IjxaEalQJ2uz10lkY6Vs1Uc/Uyo2zRqGaD/qvf06GbcAkAGPQLyaDk/teM2D9/cDYW/Qlonpd/x+1ZPdP5SiBuOCL5jEIxB+Mc5D/IXohIE8S1w0aQ+rdfv1rG5o60GWMPxwcPBClNPbvyO1+E/nBr8YcUOkrM1Z2KWTfH/a97BT+3Ubzgeuu+u1TSpAOlv7463+SQEiXqgqfe680H8llv/uklF/k3ujcjntX8F2TK3/+tPgAbj0VAPxPuwO74S82GcYpRfnGfsqa7u7rOHHvz7PTv1MNIK68rcgRGQt0LyzZ2GXTvL9mYA41H46Y7dl2q/1JuLXFlGUBkDirf1RiqvnyALEHZWzj8kbVbKZo/08SV/v2n85QELt3nMR9XDYm/9Jnc19JTuH3hdA/Bsh6RjkP3q9MrfiTXE5DVG7muCmtkY6kg3QrGzoC7J2adnPDD9sab3d2ttPbhbi8Nq616GQwmSbwvj9wTVVsv/wfs5fCh5EzoTrHJJ2//0kfbKvWRjvB4qhcJh6wwCQUK64H3yjd6ozhEc6srEtu3SKnxn+nWm93XptPwBkeAAi/BoFdpWxhduiT/lTuDylCbRpXP0itv/GxzXDfbJngPhrUTj5BPWKh+acAQAkcW1A+BcC9Exibc7CLp2mtboXtbVfQa79AJDhqYG4dSt6HpyuFTWuWfrcfSbdGRN3GQLdjfLvhPuvTNyRftQrQNxhk37bbJi/E1/Kp5MMa3YANZBmL7UBMT3F9HvBhzXK7+H9Nm5tfhpMy2Rju9ilIz9TJjZt9dp+SVauogaSbQ3EfSm2e3JabRS4xjgfR8vG70Goe9y1pb4BYpml6Vk2eztsEeb5dMm0prMaiXQrLh7694969y1R5FXu8SnLiNTmSXD0JKzNxXZrc2xsH3bptKzVSdVP+yVZuYoUJrUUJtJzs66zDVEIFw7Rg6P9fVrZCVoXuvS1h9H99+BonHIY1zNA+MUVp+k+K65d9U1kfNHPmnjonO9yci++7iQriPjUjRo+26aXw4kpLUNYlQ3DOFPR2HWmrX4c3why1mbZWGGXlv3MBLGpwMNvv0Kg/Zwh91pBrv2SpFcASO/ppYyF32uLh07/2S+uGMtufy/9tqwzOwiFI31NyT2Ws/u3xe1F9cnEAOH5VWv45b3R+YM1qB3M3GF/NECElX0292nwWviceLcNh7yZmpa1uZOXJRwb55uQjUsa26+4byA8fG4fcse3X8uJGrLfAyApfn6chb9Qa7vnvHA6r+X+fqTPq8VP+7Xxy8b1NQtzLI3nWWq1sr08yp1wVOUW7tqtzgDIZCoMkJ+HHSCtSrHkDtpQyiMYvpSALdyVmQV77/ypG6fOv3djEADpvCF4aQoAyRwgo/N2EBvbkNrs5oSEUq6ftDZykqvPpLUfSPfrcleQuut7oiS31ykAMiEA8UchSGOOIX2RHH1EAySbUa6bVinPYnaGfx5eSNk3QC68vEEvX/gv+sQI74kqaiFMvz0IByzkbiMpu15m0CMQvhbE3/ymbcTE2D+SbP2AEciEAMSHiKZVPsS5MBl3Euf1bupz7yedWh4EQPhnlsvRUBPmpww23gZAxgQgPkRwLky24lva9TItP6gRSIfvvyIz3QyApAaQ/M6SaTIcbQklUdzaD7HQ65gAklX6AoB0BAg9HLWjLaEhGKF0WPsRXOsxSIBklb4AIDEAgaCsNUiAZJW+ACAACDTmAMkyfQFAABBozAGSZfoCgAAg0JgDJMv0BQABQKAxBoiXvmS6wTcAAoBAYwoQsdOaVrme5e8AQAAQaEwBwvc7yWozJgAEAIHGHCCDEAACgEAACAACgEAACAACgEAACAACgEAQAAKAQBAAAoAAIBAAAoAAINAEAkT+bNlsBIAAINBoA+SX4xQAAoBASGGQwgAgEAACgAAgEAAyJgAR59AAIAAINGiAjMb5y930J+03nxC7+DXaFACBABAABACBABAABACBABAABACBoLQAwg80294WB1dNEW2f6BZbEnHcDFaaijsMTTYOAAFAoBEGyIHzYBvGpdliUfuQMfoxeMpdVOzaSnFJY8oDYQZj2oOCYc2Gz02WjQNAABBoxAFiV7VFyuW+z+XoB+dhP+gEkGvl/EVS2BO9si7OSebn+TKFnuTL1y4GRxiycQAIAAKNSQ1EGLc6AKRhVc+qRLvhoxxMnX1ApO6WLPvVJHEACAACTRBA+DkwRGw/fBKdbep5RrTv/L8rSeIAEAAEmhCAbG0ZL+jTtEPT+k74JDrufNWImqSu3uLnxcjEASAACDRBAPEe/t0oMATgcr+2WXpDJk7mOEwABACBxgUgbvrxNGptSgsMpO1WTP1tmTiZdToACAACjRdA9ruAoSkJkCYAAoBAk5fCRNYvgumNaZfPycQhhQFAoAkCiBg9MLoXVb9ogUGr3JCNk7keAAQAgcYEIFxxPo7NWkFlpOxr1cZikjgABACBhhcgP/cBkEifRqOuz3A3KSvUrvpuUm5ZrxWY87f2yK9ryMZJAOS6A5B/o00BEGigAEm2Ixl/yPmCN2sl/w4HCM1c/tJdANe+qG69yDYUhT1x17XQybWytpojeqxV7cVe4jrJ+Yx6J1s9AAJBQwAQPmJwHu698I7pzsO7EZnKlOfebcUz9ZuyaZ/rJw4AAUCgoQNI/t6SabKY81iUYbrebRLbCLRd55qT8gAgAAh0DADRiH6MPoslt6fXGzPDdL083Yk7NyZuFASAQBAEASAQBAEgEAQNrf4PYNc0WuWjkg8AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
⇒ 5x + 210000 -7x = 180000
⇒ 210000 -2x = 180000
⇒ 2x = 210000 – 180000
⇒ 2x = 30000
⇒ x = 15000
Therefore, in order to obtain an annual total interest of Rs. 1800, the trust fund should invest Rs. 15000 in the first bond and the remaining Rs. 15000 in the second bond.
Question 34.A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:
Rs. 2000
Answer:(a) Let Rs x be invested in the first bond.
Then, the sum of money invested in the second bond will be Rs (30000 – x).
It is given that the first bond pays 5% interest per year, and the second bond pays 7% interest per year.
Thus, in order to obtain an annual total interest of Rs 1800, we get:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ 5x + 210000 -7x = 200000
⇒ 210000 -2x = 200000
⇒ 2x = 210000 – 200000
⇒ 2x = 10000
⇒ x = 5000
Therefore, in order to obtain an annual total interest of Rs 2000, the trust fund should invest Rs 5000 in the first bond and the remaining Rs 25000 in the second bond.
Question 35.The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively.
Answer:It is given that the bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books.
Number of chemistry book = 10 × 12
= 120
Number of physics book = 8 × 12
= 96
Number of economics book = 10 × 12
= 120
Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively.
Let A be the matrix of books per subject.4
A=![](data:image/png;base64,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)
Let B denotes selling price of the books.
![](data:image/png;base64,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)
The total amount of money = number of books × selling price
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL8AAABlCAQAAAC2RvrEAAAGp0lEQVR42u2ca2wUVRTHf7vb0i7l1SKoIC8VgyFUHrGhgiagEmJCiQEa0ARBs0oEwaY8C1pQDBUQ1GgKfCjBiApxE0WLooCoqSRA0URoiARfkZbURzQVS8Dk+mGntzPbne5Md3bYyDn7ZebO7OR/frN777l3zhwQExO71izMWZTlc5ysLl8tEnctRUQQC/6Mxp8K8GQ3Q/ALfsEvJvgFv5jgF/xigl/wC37Bn7n4RxNhGStZxf1x7etZyWo2MUrwpwv/VKYZW0FWsli3T+JL8gEo4AsmCP504O/BKwT0Xg6HGGC0N1Ci2x/kDL0Ev/f4i9hv2Y9yDwCltDBQtw6khVLB7z3+CfzNvXovxGFuA+BNvqeHbu/Jj+wS/N7j70sjF1lHLgDT2EEACHGUM0Zb2xWOEhL83g+9C7iC4iRFzGAr3RN+I4vjnCUs+NMReM6gGYXiPQ24gMYO+BspEPzpwD+FlzmAQnGSQqNLauqAv4m+gt9r/AGe5QVCBJhJE4pfGGoMtSfi8P+UMPQU/Cnhn877ZBvb+byBYjuQS4Olrw/zHQ2moVjwe4I/yCGmWvZrqCebAAc5R55uz+Mcn5mmZ4LfE/w5nI5bzynia7KBKv7ket16Ay1slLjfa/wBPrL8+qGIVwEo5BJ36tbxXDIGZcHvad9fzD4j1o91Ps8z2LgxNVQb3U2A7dQk7HoEf8qB52R2UkyIEGOoYIRu70ENywkTZjk1pgUIwe9x3B/ibsopp4hg3JGRlFHGyE6uLfhTxp+KCX7BL/jFBL/gF8tk/FEUyubRwf/LRul3TzIK/7X1V4kKfsEv+DMZ/1iOGBldsUl4hBNc4QJvM950lpOkuzYrZisVVFBlJG602xDWsZRnWBKX0kdadDnDn1hTGpIMrfiDRNjIhzSanmQG2cBiegO38DGXWWi0O0u6i9lc6ukHwE3UW8TfRx03A3AHhxlm833vdEWJMIB/+YeetmoTa3LmbzWK6q7jn880ehE1uVnCLH28D2doZZyLpLvYb/YvZuu9VdTRTUcj5yk2tj9BGRllifB7pSs5/sSanPqbEv721nY399LCTH1kNYotLpLuAHajTF3OFJThXjYH+VSvKo7kMf2U1V5tqrqS4bfT5NRfz/HvQ7HNdN8VURdJd5DDtygG6f1CFGsBmIhikcuBM1VdUSL0o5XfTQ9VzGanyam/m1Fs9hL/YBZxnT4yB8UuF0l3ba/wD7JMf2oB2IJiYpfxd01XlAgFXKTZZqKZWJNzf6tQVHmJ32o7UEx1kXQHQT634B+H4hhZRvtwHuAJVrHBQdzjha7O8dtpcu5vWvEPpJm3yHKRdAdQhqJI7z2M4hQ55HAKxaPcDkAedZR1Gb9zXZ3jt9Pk3N804g9QzRH6uEq6A+jNaaO3h2zKDfyxTmmFPmsWraZcAjf43ejqHL+dJuf+phH/XPbTRw89zpLu2qYxtcwmSC/mMRbFUULk0oBikmVM2Nkl/G50dY7fTpNzf9OGfxKv6WjBedJde69azArmkMtoFHuBIHUo0xRsEIr6JKFn6ro6j3zsNDn31/PIp23AXKbv/jAXSXcdrQTFIwC808HVU+S4xO9WV7K4P7Em5/56HvcD3EpET0XCLHaRdNfRNvKr8brafMuQPALFwSS3MHVdyfDbaXLqbxrwD6DUhGUM81wk3cXmjk2sN7a7c5rn9HX/4CHTolzyxarUdSXDb6fJqb8e4b+g3cznWFytskkuku5iS1iXNf4nOWDqdSuo1ZOXTXxlMxP1UlfyNZ/Empz6myL+Uip5id9Q7GEt5YTZHefkFWMN0FnSXaxbeJEigoRZwjZLvJDNZtaTRzcW8K7pz93RvNIVJUI+zfxgGyjYaXLm7xoUa/x63JI86a4toriLZSxkeIJjw1lIOeMcDtyp6nK23m+nyfMkQ3nalYFPuwS/b/gl0eQq4r8WTfALfsEv+AW/4Bf8gl/wC/6Mwj+UWstZUsnQR/wh9ljOkkqGvuIv4WfTWVLJ0Ff8N/I4UdNZUsnQR/xBlpNnwS+VDH3EP53xYMIvlQx9xN+Pp8CCXyoZ+oY/wFJ6x+GXSoa+4Z+ss9za8UslQ5/w57NEP+Ftxy+VDH3BH+BpU5fSjl8qGfqCv5APqNSfBs6zjkqGSCVD/2e9WRw3nSWVDK8qfqlk6DP+bnxjGm6lkqGP+PtTSTWttPI6lfQHqWTo968/kUklw6uKP7kJfsEv+MUEv+AXE/yCX0zwZw5+62t5qdyMSNy1BL/gFxMTExMTExPLJPsP/bo3kpeUUvQAAAAASUVORK5CYII=)
= [120 × 80 + 96 × 60 + 120 × 40]
= (9600 + 5760 + 4800)
= 20160
Therefore, the bookshop will receive Rs. 20160 from the sale of all these books.
Question 36.The restriction on n, k and p so that PY + WY will be defined are:
A. k = 3, p = n
B. k is arbitrary, p = 2
C. p is arbitrary, k = 3
D. k = 2, p = 3
Answer:Matrices P and Y are of the orders p × k and 3 × k respectively.
Therefore, matrix PY will be defined if k = 3.
Then, PY will be of the order p × k.
Matrices W and Y are of the orders n × 3 and 3 × k respectively.
As, the number of columns in W is equal to the number of rows in Y, Matrix WY is well defined and is of the order n × k.
Matrices PY and WY can be added only when their orders are the same.
Therefore, PY is of the order p × k and WY is of the order n × k.
Thus, we must have p = n.
Therefore, k = 3 and p = n are the restrictions on n, k and p so that
PY + WY will be defined.
Question 37.If n = p, then the order of the matrix 7X – 5Z is:
A. p × 2
B. 2 × n
C. n × 3
D. p × n
Answer:Matrix X is of the order 2 × n.
Therefore, matrix 7X is also of the same order.
Matrix Z is of order 2 × p
⇒ 2 × n
⇒ matrix 5Z is also of the same order.
Now, both the matrices 7X and 5Z are of the order 2 × n.
Thus, matrix 7X – 5Z is well- defined and is of the order 2 × n.
Le t
Find each of the following:
i) A + B
ii) A - B
iii) 3A - C
iv) AB
v) BA
Answer:
i) A + B =
ii) A - B
iii) 3A - C
iv) AB
v) BA
Question 2.
Let
Find each of the following:
A – B
Answer:
A – B =
Question 3.
Let
Find each of the following:
3A – C
Answer:
3A – C =
Question 4.
Let
Find each of the following:
AB
Answer:
AB
Question 5.
Let
Find each of the following:
BA
Answer:
BA =
Question 6.
Compute the following:
Answer:
Question 7.
Compute the following:
Answer:
.
Question 8.
Compute the following:
Answer:
Question 9.
Compute the following:
Answer:
Question 10.
Compute the indicated products.
Answer:
Question 11.
Compute the indicated products.
Answer:
Question 12.
Compute the indicated products.
Answer:
Question 13.
Compute the indicated products.
Answer:
.
.
Question 14.
Compute the indicated products.
Answer:
Question 15.
Compute the indicated products.
Answer:
Question 16.
If then compute (A + B) and (B – C). Also, verify that A + (B – C) = (A + B) – C.
Answer:
Now, A+ B
B – C
A + (B –C)
.
+ B) – C =
.
Therefore, A + (B –C) = (A + B) – C
Question 17.
If then compute 3A – 5B.
Answer:
3A – 5B
Question 18.
Simplify
Answer:
.
Question 19.
Find X and Y, if
Answer:
Now, X + Y = …(1)
Adding (1) and (2), we get,
2X =
.
Now, X + Y =
Question 20.
Find X and Y, if
Answer:
2X + 3Y …(1)
3X + 2Y …(2)
Now, multiply equation (1) by 2 and equation (2) by 3, we get,
4X + 6Y …(3)
9X + 6Y …(4)
Subtracting equation (4) from (3), we get,
(4X + 6Y) – (9X + 6Y)
Now, 2X + 3Y
Question 21.
Find X, if
Answer:
2X + Y
Question 22.
Find x and y, if
Answer:
Now, on comparing elements of these two matrices, we get,
2 + y = 5
=> y = 3
And 2x + 2 = 8
=> x = 3
Therefore, x = 3 and y = 3.
Question 23.
Solve the equation for x, y, z and t, if
Answer:
On comparing the elements of these two matrices, we get,
2x + 3 = 9
⇒ 2x = 6
⇒ x = 3
2y = 12
⇒ y = 6
2z -3 = 15
⇒ 2z = 18
⇒ z = 9
2t +6 = 18
⇒ 2t = 12
⇒ t = 6
Therefore, x = 3, y =6, z = 9 and t = 6.
Question 24.
If , find the values of x and y.
Answer:
On comparing the corresponding elements of these two matrices, we get,
2x –y = 10 and 3x + y =5
Now, adding above two equations, we get
5x = 15
⇒ x = 3
Now, 3x + y = 5
⇒ y = 5 - 3x
⇒ y = 5 - 9 = -4
Therefore, x = 3 and y = -4.
Question 25.
Given , find the values of x, y, z and w.
Answer:
On comparing the corresponding elements of these two matrices, we get,
3x = x + 4
⇒ 2x + 4
⇒ x =2
3y = 6 + x + y
⇒ 2y = 6 + x = 6 + 2 = 8
⇒ y = 4
3w = 2w + 3
⇒ w = 3
3z = -1 + z + w
⇒ 2z = -1 + w = -1 +3 = 2
⇒ z = 1
Therefore, x = 2, y = 4, z = 1 and w = 3.
Question 26.
If show that F(x) F(y) = F(x + y).
Answer:
Therefore, F(x)F(y) = F(x+y)
Question 27.
Show that
Answer:
Question 28.
Show that
Answer:
Question 29.
Find A2 – 5A + 6I, if
Answer:
A2 = A.A
Now, A2 – 5A + 6I
Question 30.
If, prove that A3 – 6A2 + 7A + 2I = 0
Answer:
A2 = A.A
Now, A3 = A2. A
Now, A3 – 6A2 + 7A + 2I
Therefore, A3 – 6A2 + 7A + 2I = 0
Question 31.
If , find k so that A2 = kA – 2I
Answer:
A2 = A.A
Now, A2 = kA – 2I
Comparing the corresponding elements, we get,
3k -2 = 1
⇒ 3k = 3
⇒ k = 1
Therefore, the value of k is 1.
Question 32.
If and I is the identity matrix of order 2, show that
Answer:
We know that I
Now, LHS = I + A
=
And on RHS = (I – A)
As we know,
cos 2θ = 1 - 2 sin2 θ
cos 2θ = 2 cos2 θ - 1
and sin2θ = 2 sinθ cosθ
As tanθ = sinθ/cosθ
=
= LHS
Hence Proved.
Question 33.
A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:
Rs 1800
Answer:
(a) Let Rs x be invested in the first bond.
Then, the sum of money invested in the second bond will be Rs. (30000 – x).
It is given that the first bond pays 5% interest per year, and the second bond pays 7% interest per year.
Thus, in order to obtain an annual total interest of Rs. 1800, we get:
⇒ 5x + 210000 -7x = 180000
⇒ 210000 -2x = 180000
⇒ 2x = 210000 – 180000
⇒ 2x = 30000
⇒ x = 15000
Therefore, in order to obtain an annual total interest of Rs. 1800, the trust fund should invest Rs. 15000 in the first bond and the remaining Rs. 15000 in the second bond.
Question 34.
A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:
Rs. 2000
Answer:
(a) Let Rs x be invested in the first bond.
Then, the sum of money invested in the second bond will be Rs (30000 – x).
It is given that the first bond pays 5% interest per year, and the second bond pays 7% interest per year.
Thus, in order to obtain an annual total interest of Rs 1800, we get:
⇒ 5x + 210000 -7x = 200000
⇒ 210000 -2x = 200000
⇒ 2x = 210000 – 200000
⇒ 2x = 10000
⇒ x = 5000
Therefore, in order to obtain an annual total interest of Rs 2000, the trust fund should invest Rs 5000 in the first bond and the remaining Rs 25000 in the second bond.
Question 35.
The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively.
Answer:
It is given that the bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books.
Number of chemistry book = 10 × 12
= 120
Number of physics book = 8 × 12
= 96
Number of economics book = 10 × 12
= 120
Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively.
A=
Let B denotes selling price of the books.
The total amount of money = number of books × selling price
= [120 × 80 + 96 × 60 + 120 × 40]
= (9600 + 5760 + 4800)
= 20160
Therefore, the bookshop will receive Rs. 20160 from the sale of all these books.
Question 36.
The restriction on n, k and p so that PY + WY will be defined are:
A. k = 3, p = n
B. k is arbitrary, p = 2
C. p is arbitrary, k = 3
D. k = 2, p = 3
Answer:
Matrices P and Y are of the orders p × k and 3 × k respectively.
Therefore, matrix PY will be defined if k = 3.
Then, PY will be of the order p × k.
Matrices W and Y are of the orders n × 3 and 3 × k respectively.
As, the number of columns in W is equal to the number of rows in Y, Matrix WY is well defined and is of the order n × k.
Matrices PY and WY can be added only when their orders are the same.
Therefore, PY is of the order p × k and WY is of the order n × k.
Thus, we must have p = n.
Therefore, k = 3 and p = n are the restrictions on n, k and p so that
PY + WY will be defined.
Question 37.
If n = p, then the order of the matrix 7X – 5Z is:
A. p × 2
B. 2 × n
C. n × 3
D. p × n
Answer:
Matrix X is of the order 2 × n.
Therefore, matrix 7X is also of the same order.
Matrix Z is of order 2 × p
⇒ 2 × n
⇒ matrix 5Z is also of the same order.
Now, both the matrices 7X and 5Z are of the order 2 × n.
Thus, matrix 7X – 5Z is well- defined and is of the order 2 × n.
Exercise 3.3
Question 1.Find the transpose of each of the following matrices:
![](data:image/png;base64,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)
Answer:We know that transpose of a matrix is obtained by interchanging the elements of the rows and columns. In other words, we can say, if ![](data:image/png;base64,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)
So, let![](data:image/png;base64,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)
Therefore, transpose of the given matrix A is denoted by A’
Hence, ![](data:image/png;base64,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)
The transpose of the given matrix is
.
Question 2.Find the transpose of each of the following matrices:
![](data:image/png;base64,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)
Answer:We know that transpose of a matrix is obtained by interchanging the elements of the rows and columns. In other words, we can say, if ![](data:image/png;base64,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)
So, let![](data:image/png;base64,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)
Therefore, transpose of the given matrix B is denoted by B’.
![](data:image/png;base64,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)
The transpose of the given matrix is
.
Question 3.Find the transpose of each of the following matrices:
![](data:image/png;base64,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)
Answer:We know that transpose of a matrix is obtained by interchanging the elements of the rows and columns. In other words, we can say, if ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQAAAAAnCAYAAADgrJZcAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAVlSURBVHja7Z09bxMxGMc9dYINJIpqtkqIrXEYizqg44RYWFCTU8RA1Ql5YchUXbt16QeArRsMzAx0R/QbINFPQMVHaIlz+N7iu9j3Zl/yHx6plZpr4ud3P9uPnyTk5OSEIBCI9QwMAgIBASAQCAjAkQjDySYj5IoQcrsY9Nrn4U5b//u9Rz+o/+8s2Pgc0PQ7bLKV4uvGFb6cTdKA0Mu2k2GcOAhgJQTgGls2+UKSIAAIAAJAkiAACAACQJIgAAgAAkCSIAAnxpuQWxYcv4IALAtAVk6p9/4DBGBfjowFYT43STWZXU3CcNNVsetwlLym5qvyEEAFARwH7NUcri3/4vD09A4E4NbzCLm/Qwm5dl0AuhzFr6eF8YYAKghgzMh5V+ekEEBx8ODpvuq8OLsKcFcAuhxFomiHNQjAUADZ2YXcdrENCMO9J/fIxt/Hb8PXriRo8njjnNzf+2H95u+pAEw4mgugpZvBRbZs8qVdjImjg22AaZJCPtp+wR59SgP2iL341OTztCmAMaNn6s616GZfEACf7A4ouZz/TgeFs52QSjJm9PppwPfLagt+4L+Lrzv7e5MCnQlH3GMHbc3QJmx1wZXTAoiWS4PvAfdfpgeh6cpsnSSJ2ULuF0ViMjNNg7KyvQKI989LVwCz2ZWNz5atCuIb8v+4yeV5emZWzdpVVhu2OKrDVldcOS0AuRQ7PT2842+Ri656lk0EkN9XZp9rc/vI/ggguinLxiENs7wJk+snN7XpdV3jqA5bXXHltADEICwCom/+0jfW5CIz81RZAcyWulUBXX0BZGdbVTFOJQXT67bFkQ22uuLKWQHMgaD+F7nUyS8z21y+VS3U8MB/OaT0Wzr5TT3PVRHAwiysiCYFYJOjpthqkytnBSAMm6/UJjNHu8s30yQJKGWhTBSyTGeodRJA9m/LZ7ImBGCTo7psdcGVkwLQmSXaPG4yqtSKApOsTCv2mRBA+QqgbHzqCsA2R3XY6oorJwVQdA67UG1e0hPQRQ2gvFgDAZTXAMpzWFcATXFkg62uuHJOAPKFFr3AzPKtpZ4AE0vnl7Nle8wImgjk9M/rJoCyQlw48p/FhcEaAnCBo6psmXKVZ0v1ey8EEO15RGKK94YLZ8N+EDadPF0B5GEU+zTORztFgPZZAPlzaM6f7zKPHyzmZPmMpTqOi866R9uex9+oc53cCEkzkFoArnBUlS1TrlZCAMrPSkuZuXQ/17DBKx0Dis40eUPMZjEJV1OdW7YFEFWjk8490exTmLfZWFBKfhVV9/MFLr1OQL3rusRRHbZKuPqzSp2mxCbQTQjAfM84s/KEHZgWoFwQAMJNtvJNVBnW+rQFcCdJ7bxjK1qWRklJ/6wtEHwgyAoIoH22VL+7yhcEAAFAABAABAABQAAQAAQAAUAAzQpgOn0IATiQpEzjyJZ/4Q/J16oCiK819L+K94vLCrqsqsvGE/G+9nRlXVTxZUU5vgYdXA6H9HNXH7YCAbQvgMy3/2iebCj5UrBRxJQuk2u9AigztkmC4qMucR5/zB9Ex1rsasR5dFacSrrshZ8f4bHx2eJx2OxxId/OPw7R7xVApQnqP1/L2FAxZcLkWgtAtqeaNmooBZDuEc8NfPq6czhyCdB5HMJ9tqrypCUAHaam07tVmFxbASSdbEmDR5sCWLoCgAB6zdacBUp+mvJURwClK4BVEYDL3w6sO9j5/ZqYLSQkEIBVtn67/O3AaTYOw8PNmI3p9GERU1pMKgqSgKKWJOi157GjuCXW847iVl2Dd0oOh8OPXX7qMqIXfN2YsJE8JmbyJsXkn6LHY8ARiDUODAICAQEgEAgIAIFArFX8A5MBBrwvIYsbAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the transpose of the given matrix C is denoted by C’.
![](data:image/png;base64,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)
The transpose of the given matrix is ![](data:image/png;base64,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)
Question 4.If
and, then verify that
(A + B)’ = A’ + B’,
Answer:![](data:image/png;base64,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)
(A+B)’ = A’+B’
Explanation: We will first calculate L.H.S i.e. (A+B)’ and then consecutively we will calculate R.H.S and verify that both are equal.
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![](data:image/png;base64,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)
Therefore, ![](data:image/png;base64,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)
Now,![](data:image/png;base64,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)
So,![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
From equation 1 & 2 we verify that
(A+B)’ = A’+B’. Hence verified.
Question 5.If
and, then verify that
(A – B)’ = A’– B’
Answer:We will first calculate L.H.S i.e. (A+B)’ and then consecutively we will calculate R.H.S and verify that both are equal.
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AwRwghBMwRQgi50P8BHt++CKg6bpwAAAAASUVORK5CYII=)
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xQKfCJIo26dhSwgOei7u11gA/wmUaeFIdlLoget54nXKTNIlPUbPiorXR0BZney81fVx0i3ldN9o/quqeg8MG9HWzzU++uBjb4FJXSAD6LDx8FmZbcPFbFMZH9qM+qGCyZ80qq7bBYqJKox7CHC6d9T1X1fQB8gM9UTn5Tto5TJdai88L1/Nt1QScsU71whDjdcL3dJqT9rAsJC4iAgM/iw2fo8L80+4YCUZWb8qbaHw4iowrRCuzhN4s9XBoBAR/ggxog4LP48EFT93ngA3wQ8AE+CPigmQ2Lw+QWVWpyuAlFHatgIxwm1wypeSrbGikeDvBBwAf4IOCDgA/wQcAH+CDgg4AP8EHABxnwGRpWFYdYIeCzyPDBRoAPKtSw1A66tg0Rm7NAc9UhM27DSuBTv42UsfknkBm/SattPRAPDSGEUOXiISCEEAI+CCGEll//Bwe58oUOuByDAAAAAElFTkSuQmCC)
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASsAAABgCAYAAABFTFSvAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAuASURBVHja7Z2/bxtHFsf36ou7BEgED6uzi3Tmkr2LgN5zfNf5YmqhzlBB6BawY4CFLke7My5R+rg6d0mRNjnALgMcrL8gBuTyKvlvsI4zu7Na7i8Olzu7Q+FTfAFDsrjDtzOfee/Nj+c9e/bMQwgh14UREELACiGEgBXa/GV73h+qhH26tTm2AVaoQovFwWe+571bDpQPRYnzIFrcwk7tKgrE43J7exdicvQ1NgJWqAJWQ2/46mCx+Ax79KujifgaWAEr1BKsXApbXA2bmtoIWAEr1BKsFtH0hi+8N2nYIoan92fHN/vyQjzPf+eaR7iNjYAVsEItwGoRBbd8P/j28Pnzj1YHZffAkG0RnvfeNVjJdo1Gwb9KbTSf7wErYIUsw+r588OPJv7kGz0I89Dww6f3Om2v8E5lMtolWEkbBX7wj21sBKyAFWopDKzycLqE1b4vTkQQLoLr3msXw8AKG50DK2CFeoKVTBzv+95Lz99/2VVbn4b+PeHvnygvZgdgtamNgBWwQi3CSg7A2ez+UIVio+Dnw/n8WlftHInRj7KdrsOqqY2AFbBCLcFKQUJ4r/QGRrWJcenp5PM0tsI/vUHVZViV2mi4/70JsIAVsEIth4FyQEbh+IHMxXSx6zo/iLuEVd1RpLq9VCU2egKsgBXqGFbp33awhUA9Y+m9ZcEwnx9eS2E1n+/Z2hxacyymoCrI6AS7iY2AFbBClmAVh2feS5uwijd/lp+jS7T8t3/mcqLd1EbAClghi7BSMLkevO4ib9VHGNgacA1sBKyAFWoBVmW5GQ2NrgeYq7Da1kbAClihLWGljo3Ia2TUObfZUA5IOQjVBk1//6TrNrsIq8RGZ9vYCFgBK7QlrBQcRoMfVhLKYvRrEEZ3+2jz5fYAd3JVbdgIWAEr1EIYiOwLWAErBKyAFbBCwAoBK2CFgBWwQsAKASvXJG+V6PIWC2CFgBUCVsAKAStghYAVsMIewApYIWCFgBWwQsAKWAErBKwQsAJWCFghYAWsELACVsAKASsErIAVAlbAClsAKwSsgBWwcqbhliqdXNX2AitgBax6GNDyClnPE+91Mcwu2rVpPbk+29smrPqA7Kb27bO/NmkjsGpmR2carMoYGVQG6aKGXVZllXcL5aKEeKsqFh8W29N1e9uCVRRNb00m/jdCeG+7vM5EtlPdA39p64u+QV+m49n9m6psvG6jGJ5OpkdfAqsN33c0vVFmxzJoOZNPiTvo+k6pB3/X9wGpwa6Mugqdp1H0aew5lQ+qvtq7DazUYBoMfpeg6qISc1al9QMdG9iXBU2L9Q396eIvwMqOHZ3IxSQd9MJkYKgX3YOXsq7SSvodct5hX+1tIwyM294drNSkNbjz4nA+v+bqANOetgjChX7P0jtIvO8Lk3cNrAztOJ/vtQarNgaihoB/cPBQeVdrQsG+bllcB6uqcM+VWyF3AVZxRePl88To16pQwAVvwJ9ED6v7x/rowAas5GeqVERD0BvljAxyc9nf1/3/WjsqYIn3X8yOh5WwMkkk5xWF4wdCBD81NZIa5PLvl4CKO2v9y44mk7/14aWsg5Ue2HnY9tXeXYNVLiRI8xf52dVlmZaPtwGrdJAvP3eTsVgoLabqIR7fzOeV7viDF2npsSUUj6bjaRY28nPCYPxIpg5kCBeFwV1feG/U/x/uf79Jm0Lf+3etZ1URP5qr5EuavmA9GNIB76CLvAKrpRELwFZelTj3w6f3tp3Zmq467rpnJTWbzf6kOn3cF0vDAWfDGtk/DPqurTAwHcOGwJJ9YiTEL7rmYVrANWPzJJ98pry25SScWXBa6RcxYOKJZuDfeaE/MwrEY/lzk1xeHrr5/t6uG+qJt3nXbZMB5MrKWT2sSlYDk0G1yXcvyXcZTQpNwbFY3P78Y+/j/91eLD53GVbFcMDNyascFP4bowWM25889j65/V8b7dBwMHEeEkfhSXYy1LlX+fPUWyzNwxb7ov7bLJg0QPXnGduxZIJq1VBp584kzdYO0oSgUvP54TUNhKYeii0vpSoMlD9PPCvlCfRRZv0qeVb5NqczvcXJq43+IleETfusiWfVJCWjpXN/MqVStZ0iM/mWT4zL9lWtZDeBVZmnVB4Ciu+q7Nj6i08bvObFrTFW49nUlpdinmDfDrTAqux92ttnlXoiW/SXTRdR1sEqt7eviS6yewLL+qPeLlTX7ioo2YLVOju27lnJUHAchI/WeVbqC5es/LkaCq6D1WWCtRloyVnVhQXDU1ePCUkbla1q9ZGzKvRDT5zr3FEVrOra0SWsTOzYmuupQCXEb6YzoDRoGfGzuSGXPBSbsOomZ7WbsFJtEMErk7RCHyD1/fCfjb6TJVilfWnNFqCVEDuXH1K/88NFBkpPbMLK1I7FlYSmLqcwSy6azJYurgqug9V8frCnj4m4WG13V2Elc0GuHbVJowh//yQ/actjOL4ffFsHik5WAw3gnqzgXehkfDbJLh2FFGgSfJnVxTZhVWVHtTKZs+P2Mb/eZ2U486VJ9FHwc9Xy6vHsi6HO/4zD6EHfHTNO/h/s6eM22Rcbb10I7ibnmz6YnG/cBVhpb1lPGrY3aEooySVvbVs5OcifufD+S/t80j/LZHQKw+I+q036ROxd5b+Df6b7ic7pZfdKRRP/Yf57ZhP7eoUxC7Y88JrYcfvZRYx+MfaoLo2TeGXLmDo3a6ZfzpGzYabJThkCy4HlIqg2hVWVl23Tw8qsqKpnSXBNo8UNF0FV3xeKfboLWMUeSvjdpv1PejB/FuLHrN3znyFX6PR3k/noxAYpTMrGiPxdIb2RAdbyM75aZ8faHezo6or7rNwKIXf5bGBVGGg9wqHzACsErBrC6kmXz6XzACsErDYLhZOcVV3eGVghYAWs+gVVfhNthwtKdB5ghYDVTggjACsErIAVAlYIWAErBKyAFbBCwAoBK2CFgBWwAlbACgErYAWsELBCwApYIWAFrBCwQsAKWAErBKwQsAJWCFgBKwSsgBUCVsAKXRFYtVFVp1Gn7Om5XbUJWAEr1BKs5D358v7z7DXDqny45buLis8V78fTo2nftpN37PvCe6MrHTetvA2sgBUygpVZ+fi0is/14LW+A13e063u2LZ4d1H2ufoZ+r7zrm+kLIAlU39PgkvCNFvBZVP93f/jiSf++h/6JbBCW8AqGZyFCsi6+KytWo5xcYFimTNdkXlbb6apN1pWtThta67eHrACVqhDWFWVa9cD1was6qoDpxW6ewibKm2RtKlpwQRgBaxQC7DKwqFQ2NJSZeQ6IJWFh12o7rkmpdeBFbBClmElla/Uq+tD2qrjlw7+EjCsq4RtOwSsbVNDgAIrYIVagpWUXJVLK+NaXgm8BFIxN1UHDav2MvH2GgIUWAEr1CKs1GAV4rdkG4F1YF1W4vbfZUvIh4FY6BL2ndoLWAEr5D6sZGJZiOAn6cmk2xYkMCx7Nyv7mTxxLgEZBuNH8tm2ViHXhoFlSf8tvT1gBaxQC7BKPJwVOCgPJ8ljdVkyvK981cqzSbADK+QmrPZ972XZcn06eDsMx+KtA96Hrr2q1ecX91Ntu+cMWAEr1AKsEkBc5Hdodwkrma/SoBqH0YPebBZD6Ty/gz6B2FlTbw9YASvUAqySEOdMeleT6dGXEhzyzF6c6G4+QE0AJZ8j81ZD4Z2qIy45764PxeXSxblcpZRtlEeA1HGbLbw9YAWsUAuwij2K6Y1gNPhBH2KWGvh3XtjaZ6W8tjiJ/8EbDH4fB+GjLrcprAVWAih9kHlbOwArYIVaghWyK2AFrBCwAlbACu0+rLgp1BWpfWwdbgEBVghYIWAFrBCwAlYIWAEr7AGsgBVyHlbz+V62+IFrhRmu3AArsTWwAlZoDazizZ6Xe6cu5cYGzKumeENpmb3VvfLAClghhIAVQggBK4QQqtb/AeIEKC50roT1AAAAAElFTkSuQmCC)
From equation 1 & 2 we verify that
(A-B)’ = A’-B’. Hence verified.
Question 6.If
and, then verify that
(A + B). = A’ + B’
Answer:![](data:image/png;base64,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)
(A+B)’ = A’+B’
Explanation: Calculate matrix A by taking transpose of A’ and also find the transpose of matrix B and the solve L.H.S and R.H.S separately and then compare the result.
So, A = transpose of A’
![](data:image/png;base64,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)
B’ = transpose of B
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
From equation 1 & 2 we verify that
(A+B)’ = A’+B’. Hence verified.
Question 7.If
and, then verify that
(A – B)’ = A’– B’ ![](data:image/png;base64,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)
Answer:(A-B)’ = A’-B’
Explanation: Calculate matrix A by taking transpose of A’ and also find the transpose of matrix B and the solve L.H.S and R.H.S separately and then compare the result.
![](data:image/png;base64,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)
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From equation 1 & 2 we verify that
(A - B)’ = A’ - B’. Hence verified.
Question 8.If
and
, then find (A + 2B)’
Answer:![](data:image/png;base64,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)
Explanation: Whenever a constant term is multiplied with a matrix then it implies that every elements in each rows and columns are to be multiplied with that constant term. So in order to solve this question we have to multiplied every element of the matrix B with constant term that is 2.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, (A+2B)’ = transpose of A+2B
![](data:image/png;base64,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)
Question 9.For the matrices A and B, verify that (AB)’ = B’A’, where
(i) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Explanation: The product of two matrices is defined or possible only if the number of columns of the former matrix is equal to number of rows of the latter matrix.
![](data:image/png;base64,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)
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PZNmAnw+S0wXqZNuCjpN5yL87XJvtuZ+TLGkbXdsn40HniQ0tlR72AnsfQix2h9XJqkoB0frF1xl2cJ6w2tJwktosnO1LoebROz2UbMCiMr/9ZPUlEznwJ+zN2+WzbLokPfdTxIQPo/QvQswK9Zp2nZ3NHhoFze78jA2q22EoE5mOAHqBnde+tjsTeW5ykDAF6gJ67kCVnssO2eHitmPWDIEAP0LM42nI/e2rj8FIIAvQAPe3RnusDKzlsMcqDAD1Ar5bnZrk9ViOjxLapsqBlC1SRy4MAPUCvVvBRGo2cOGGXbiKshQC9qwS94fC6zkkkPoIPdW+hpkp1AhA/xAHQswc9vmrd84WmEHRVR3Sqxd6AHgRBGBnCCBAEAXoQBEFXVP8H0I5/EDdYJOEAAAAASUVORK5CYII=)
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)
![](data:image/png;base64,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)
From equation 1 & 2 we verify that
(AB)’ = B’A’. Hence verified.
Question 10.For the matrices A and B, verify that (AB)’ = B’A’, where
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Explanation: The product of two matrices is defined or possible only if the number of columns of the former matrix is equal to number of rows of the latter matrix.
![](data:image/png;base64,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)
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SAw+HFspMNrZtMCZ0AQAiqEICMkBN9vg06LvONi1WX5d3hUwAQgiAIAIQgCAIAIQiCaqv/Azy4sDK+ffErAAAAAElFTkSuQmCC)
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![](data:image/png;base64,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)
From equation 1 & 2 we verify that
(AB)’ = B’A’. Hence verified.
Question 11.If
, then verify that A’ A = I
Answer:We know A’ can be calculated by taking the transpose of the given matrix A.
![](data:image/png;base64,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)
Now multiply A and A’. So,
![](data:image/png;base64,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)
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)
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)
![](data:image/png;base64,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)
And we know ‘I’ represents an identity matrix
![](data:image/png;base64,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)
From equation 1 & 2 we can say that
AA’ = I
AA’ = I. Hence verified.
Question 12.If
, then verify that A’ A = I
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAhkAAABDCAYAAAAxkBczAAAAAXNSR0ICQMB9xQAAFUJJREFUeNrtXb9zHMeV7mV6kpw4OMnojU4KHJnbQGjZAW8x+kFlMDHYYiK6UKot1JYpSrV1BVMLZqySpUtJKhDFwPZd1YUnl0kqc0DyLxCPYHiBAfwJBuHqnn4zPT3dMz2zswss9gu+KnIx0/Pe6zf9vul+/ZrdunWLAQAAAAAAtA0YAQAAAAAAkAwAAAAAAEAyANPIjHV8gH0AAACA8xDTQDJOAZPJ1TcFYy8ZY6+K4IfRaHIRdgIAAAAWATsRv95h7B+umMb7n3zO2EkHJGPOJKPHeo+uTiZvwh7tsWfYArYAAODsYBjxT3l/5wZIBkjGQuL27e3XIs4eJYxZvLw6Hr8FW8AWAACAZIBkgGRMjVG0do2WlgaC3Wcr0ePt27dfgy2W2xYAAIBkgGQsOcloe1p/MoouctZ7BuIGWyyj/wPAWfQrkAyQjFPDluBfMcaP2kpyTZJpxUMEVthiETAQ/A8yyfvScLcHewAz8CsBkgGSsdQkg16G1kjGKLoo+qPfwr9gC5AMYIk/3r5kjB+AZIBkICej7UE7iq4hBwG2AAAAORkgGSAZ7e7R7vMbYrD3IWwBWwAAAJIBkrEEJKOy4pvnb+ZvIW3IoMr7O5/Jf4/HV9/q90e/CZGlboKUfs6NaexQ9ryq686SLYCz4f9t9Gtb/jkveYu7qiiINfffOhWXp31Pp7XDvP1q2r4CyQDJmAnG4+3X10X3nqrZwMXTeBi/K8RgQn/fHW68I//OGX9OORnynkG0dp1z9lx+iY8G0fuCs6eqYlxv62vX9L/aqllRLVU+q8fZM5JFroGrmhIr7LGqJxHYB01Ihnz2Kuffk3xdsf7N9nj8um0rpTdjh6SD0te4rsqe87YFUO3/0Wr3bmLn3jPqLxqUTf+nnIyc/8eTywX/t/wmn4MTv03Xyuel/arqpvhrprj8c2N7+w2Pfx5ovzqS8tjXqeclOh+7dK6QV4TIOwuSIWVZ7Xb/V8mtbPDvBRsYdvhd0Q7jN0ps8CufDQw7PLHt8B5nD6UdNsbjnzlsfCdp/yK1/wW1L9uTz++y7o+UkyHvkf3X7bIfL8Y3P9J+9aSj/OrKf5ryV8ioninbVeMRZw87DhlBMkAy5pR8xL6jgKycXAZAsfVdmpTI2D5j7MTcXWIEyZOuWL8XDUbvU3CXvzVZAkjLt+uaEWpA4at/2tjd6Jm/z4JkJNtI2eHaYLRp/t98ZhrgefSIfhsN1q4oO8jrdGAps+dp2AKoSupk9ynwmf0l/6/9/wX5P5GM2OP/MojK3yXxKOnXffIXq1/3TT9y+me8E0u59m5Ev1AB1Lheyr4uA7/0T/2b9s9jttL/wQzGts5bgn1LOrch7yxIhrbBgcMGOd3IDh2+nrNDJ7VDEqhjhw0uiPg7O8AadnjBVtYfy2dpO/xxY3dDmL+b9yTtDx3t//cFKbu8r6MJEJEMeU0n86tv+vHOB/J+Gfyl/JJ4eEmQilPsRccjY/J7OUkByQDJmNH2SfbSHBTVb2LwRY6Rq6/n/BZWF6HQA8FRk6UKGZzt9vYG4sPV1dW7ddusQzJS/SwikOwoEPvU74m+xW28ZAe59EGDcpk9520LoNL/VX+lX5i6v+j/2Vd7RjJ8hILIAC2D+WbzzHv2rooPjH4t3Jc+3yIBMfmnnknQ/lnYAUNyUtuksxmwbJ1d8tLfDHm9erZNMjIbxDkbJLszxAvzCz3Rt2gHHahf8f7wc7k8KQNviA2IMMh7zeu1He647ECkpNh+nPOr99QMQ0YySP7CszShSp5z0vGMGd/a901yMg6994JkgGTMDOYASqzZH/TCSUbdL3cfkcl+L58dsddezXyHqjVNkrms/VQOxwxCqvNK9Hi0N/rXEHs2tkVSivwEyaLVPhByymSo/yfBNpxkuGYFfGQl16+OGRBq0zc7kmvDMbOQm5Ubj1/Ppvj5Uf/K8EOfzgHy5shHSJ8Ms/fyQp1cA9MGZdebdrBnFrKZj/4P25PtN0NsYJIBu55F9ns+sBuzKYpA6PYv+IlBvt0yksF6mw/kTIhLRv081VaRyBRlBMkAyZgbRoO1TZr6ZZw/p+nIeZKM7KTbYq6B69muL7UQuIKznCHwDfC+5QvnlLKWPcSes7LFskL7wHGID9hf0umSQkl/tUEycn5i5TG42rf881WAfzqXL1zPtXUW8c5WA3lL64YMa/RJFXmQswYsIFBmyxpFkpH+TecmuG1QslTiyGlIZjjcxbR28u3/n6v9tkhGmYzmM0AyQDJOsSBUkoSUBkeLJJwWyWia6FhnuYRIhvzCChrELZLhkrFgz97Wg1CiYZOWaW0BNPB/NZhn/TVLklGVREkBtmxpooxk+No3dD7OdD7puAJXHXlnsVyibXAslzrCSEYxR8KQe58CsbbBnbwN8kHcF8DTHBijvYI8N67+W1n7syYZ2QyHX0aQDJCMuQ+2OpM8Rx7mt1ySD6BN26tDMtLAUDJo+qaOcwTJIaPPnmFLWHZQaGYLoKH/G+Sh3eUSZ78e+oio6Z++QBHgn/sumawdCSfmjIIpr/ncKnlnQjIoyLJe6Q4JcynInl1IA7EjudPcOeJa+kja7Lns4A38Ntlwtd/mckmyLNJcRpAMkIyZ5WT0Rf+m+fXjylGYNckwksxygzs9o+5uitqJn3pNXA6c0ha0VixnOUg/O4GusF482PuwxJ6HdfIoSmzxqm5GP9DI/w+rSAb1h5NkeAKwK/Gzql+NtX/1JSy/0H3+qRMbXf55QARC+buIfu/TOZ9YWUwmzORdr+2H0yR+Uv6DzwbUZpbg+cnnuZkZww5q27LbBgeupRuXHdLlIMeuDbKxOZviat+13OIiGSFkwSejvasGJAMk41QSP7nY+oqCazKIZLsqdCb2S/NLR4J2QJhBdxKLyykpqDkA2Qlqu8NLPfV/Hj2SB7TJAV4OKFE8ebftLay07p1sVSTwo8IWVp3wRuv2arBSWe+D+6H2nLctgIb+r2cbtP+rbdw0gEvEif/nSGfqR9aW0cKXpZ7OT/t1JXpc1q+0XOD0T2MLKyX56RyLC5l/Zrsy6DqpMwXroaVzMVkyJ6/6fxM/nGYLa4gN8ksEeTvI38gOdA0XcWYDRRryO1U8dviJ2Y9xYgdBdihrv2O0TztcEr/6j49kYNd+dT8jSQmhuJn4VSlZkDJ2Gfu79r1KGUEyArLIQTJaGmSjwbXR6NIvs4I74mk8mrydn6o1EueiwYSCrZlMl37hEIx6EnWCayqHbFcO/qpOhJYjsE0azKZ5tqx/QHYw7ZVsbc2KaK1Fg+s5IhINPvbZ8zRsAVT6f6G/Noa77+RIQbj/Z4mOni993a9Pcv2qalBkQcF1n6qtYNwn/ZPkdPjncc4/rSBs6JwWl7Lb8j1XBs4QecuDWLPxu2iD/j2X3EldCrm9NbODiOJPTUJWtMFFrw2cduhtqqJrWXDP8kAc7R+b7af3GD60Em3uqfoehg/JWSlJADuWX7kIrCXjsV9G/4zGmScZ0wT/Ove2few4cjIAAACAZceZJxlq+rxhJcLQe9M1/zlm2INkAAAAAOcdaklKDO7beR9nqnJkkxmGOvdOU1ESJAMAAAAA5kwy2shvMNbjT+oSgDr3JolV860TAJIBAAAAgGQ0aTQJ2vt1i6q4ahyIq1d/W/fwprr3Nj3CGyQDAAAAAKYkGaH1+3PnPMRrMefRfzXde6+WMOT9t2+/Vrfscd17R/3+b+Yd7EEyAAAAgKUnGa7thrVQso2pKmmTZheoRHNoMaZp7g3ZrVLnsCSQDAAAAAAkY0aJnzLId1n3x7JDb6oCcJ3dH9PcG5jjEXRYEkgGAAAAAJIxh90lVLFuJdraC8mrUAFd18RPKyAGHM097b1YLgEAAACABdzCmtbkrwj02aFWzlmC0mWPae4FyQAAAACABZ3JkEsmZqnk0lkPx26QkGWPae5FTgYAAAAALMjuEvM0O87530J3hsikTddshzFL4V32mOZe5GQAAAAAwGLtLjmRBwOFBtLkNL7es9KZCs+yxzT3YrkEAAAAABZouWQ0WNvk/L0/hxbQShM0V6P/8dXW0EcCH0myQMdiT3svSAYAAMD0qLtkDP1BMqbKweB89fvQAKrPGdk3j7q2l1doh0oOYnB/mntBMgAAAKaHXJJ+j7OHyfgqXm6Mxz9bNv3XE/2PGestnf5nYncJAJIBAMD5xChau0Yfd1uCfasS8BtWe15E7Fj6d1bWH29sb78B3wDJAMkAAABoc2xTeXzi6TRnVy10UL0R/ULqj9kMkAyQjAVe94TeAHB2xzbBxMNFIhltvl9a/78uK8kCyQDJWGhsCf5VncPyoDcAzH8mYy0aXVskUjwQ/A8yn+/ScFcso/4gGeeSZPz65z9lP/3/X08mP4c96g8GyxZsl1VvYAF9NYo+XrR8jC3Bv2SMH9Q5Y+s86T/TfBXxL192+OW/gGSAZAAAAEyZAMo/lQUPl/Urfrjk+oNkgGQAAADMJKdh2Oc3eH/nM8bYhfH46ltRNNpYlmCb6T9cSv1BMpY8JyP0bJWQ6+oUnHFdO23bvr+7njELGb0vUjLA3mijj+ahd1PdT8uX2uyrZUwYnnV/qW2rqkZEVq/ItfTQRh/K2RL5rk3jN22/X/PUfxFnd5L+OumAZJxDkrE73HhnlfO/kPN3xfo9e71QVkwdRGvXs/Lx/JCLra9c162L7j11DRdP42H8rhCDie+5Pc6e0bXyhcvOkCkeVBfStmxTXsMZf065CansnD0X8eTyaBC9Lzh7qs6Q6W19XVZxtq6MsyIZVbo30rtiTbiJ7tqXvjd86ZsgX3LIo6r0rnbvhvgSPZv0Y7z3LJWXs0dKXmTys5D+sms3GP11oPvrSPaX6zrdX8fS/tRfdYLjZBS/bfWhaNKHdUiGaj8v969suaVc8hr5fhExkPdJu3S77Efj/XrSUe/X5tfb49u1cy60/k9s/ZMiZue3gBlIxjkmGVRKfW0w2rxlnkZrnFSbBhYePaLfZEl49SJYJ9rKA+gokKqXULD7rrNg5FShYOwl3S9frlW++ueN3Y2e+bv1JVDatpRdV3Y9MXdZqOv0OTmSQEWD0fu3sgPtvAfi6Uqxtox/KpNxViSjTPe6essBWOkdTy6Xb7GrpzudYWT50qFZdMnjS1dSXzKIhk9nV+BIq/rqNix595et8FN4rQp2SEco6P8f2P21LgO87C/9m+6vY7bS/8EkGrJ/KLDL+9SXu6e/SnyulT6sQzJccl8QcSq3rGkhGHvR0QSLdpeoglrZ+/WNfL/UkkjEP5W/X4xvflSLYCX6v2C6SJfW/48buxvC/B0kA1gYkpEO+BYJSHYpiH16bhKMi1sjKUhTIKDAZAYv9ZsYfOEKmnaAl6XdV1dX70qSYwfj0LazL+28vC5CQYTKF/jryjgrkkGDb5nuberdRPeavlTYAUPER67X2zrTQO3zpZy8ho3ktrimfXXekc4OWCQgpv7SMwbUX/a0vu6vVzq/okP9ZQZW6q/QQEuk2Oxzow8P6VltkgwK7FVyZ7MJGckw7ZDzU03W6sib2J7dt8mJ1v9OXf1BMoDWSYa9XuqD40vmqOxo+zR4+L9c01kPM9D1450PfC+ELyBmvxdnF7Ip0/K2s4ATHmxdMy1NZKzqk50s6a1TJ9cgVPc29K7UXclR1L22L1lfpPasR53+NuU1g2FO3pJZm2WexSizS2q/wP6iINy/Mqy9c8Lsb08f5gJ51btmJpiWvWt15E5IUL5ORhnJYL3NB3bQDNA/17553kvdmRGQDKA1kkGOHgLzi46Oti8LDPa0ufNvxvp8uowip+45f+46zdZ1X1mgTPUMaLutYNtUxiZ9UkVWcksKJbq3RTKa6J76UvUSjHPaOzvEMFt3D9HZd68VGI7aqG1wroof6QMhZ9Bfx9RfIt7Zqr1U4u/DQ18fDpN37TjoXXMQFbfcRXIwS5KRLpU48i6SGY52ioCBZABzXS6hwFA2lVxGMnxJgJQklQYIK5j5glhIQmVV27MmGU2TPtvYXVLQvbf1wBwwWyYZ+3V0p6BFyx11g1aubSPIVOlcFqCQ9FlRYdFY7mjUXw7bGv2VBO3AQNt2H9bdXaLlvpOXO7+d8jRIRpoTw8T+eT7nBCTjnJKMNNiUvMC+qfNcIPYFqyxTPBfk/OSkPPiFtN3+cklzGdsmGVW6t7pc4gwgJaSMps+rfMkxJZ4LMr7ETlNn6+u7Qt7DOsmHy7ZcUrZjoYX+ekL9VWV/sw9NeVI5HeSyTZLh2t1hyz3r5ZJkWaTn0v/ARXpAMoDFSPzUa9ZysJBfK7R2Kb9MKTAZCZ6fuXam0HWyvb7o3zS/enxr9TrJKxcw6Dm+WZPQttsKtnVlnBXJCNW9VG8zaTSAKJXo/qr8yzaZbfD5kp0wWMgRqPalQ9fyUl15kfhp9tfmA7lrwddfaktmUkCq48vpKO2vAJJh7tYwcw+yPlyv1YehJEPZQUS/d8h9YMsdl5CMXOJoA5Lh0z9dClI7S8ZvgGQAC7eFlaa5k+2PBH5kbjE0ByRaF1d74tVugsF9++uf6mdQsqO5u8CXOKYJy6F8rjzoS345SdmiePKuKUNV2+nWWOMLSoJ2H5jBbRKLyylhcAxgdWWcJcmo0r1NvZvq7vUlezu0TuTL+ZLa6SAPSEpkLNXZMVNiyisDZirvSvS4SV8t0ZLJsbO/jC2s8gu7o77WVY7Fhay/sm2eZn8RYRmW9JdTHnX8udxCu272ofp/3T6sQzKkflzEmdwqsIsX5oyCfr/UNnEiARKx3hHD+598TgFSvwd6i284MbD0/0m2pXj9cZzoL0j/8zYzB5JxzotxqToLVABHFuRZje7Go8nb9suYbEfMqtWtRYPr5te8GmiiwbXR6NIvs4I64qndlu+5KqComgw6YNhEp6LtbArYSHSNBhMKambya/qFRDCe1VTG0IGvEcmIBh/7dJ+F3k11L/iSWL83hS99HOpLAfIemLUegJzNnpj9tTHcfcfTX8e5/jJsafVXWkzKbisk0JryyOCf1MzIAm5IHw7rzGRUyJ0Gf+O9WYk295JciSzhVM72JMTKSEKtOQNT0F8V9Rq/Ln/vMvb381orAyRjCcqKAwAAAABIBkgGAAAAAIBkACAZAAAAAACSAZIBAAAAACAZIBkAAAAAAJIBtEEyxuO3qs4hAQAAAICzDlcsA8k4RZKh92Y76vAXT7IEAAAAgLOKnYhf7zD2D/e5WrLgG0gGAAAAAADzmPWAEQAAAAAAAMkAAAAAAGBh8E/mLnYQklO03wAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
And we know ‘I’ represents an identity matrix
From equation 1 & 2 we can say that
AA’ = I
AA’ = I. Hence verified.
Question 13.Show that the matrix
is a symmetric matrix.
Answer:A matrix is aid to be symmetric only if the transpose of matrix and the matrix itself are equal or same. This means that A = A’.
![](data:image/png;base64,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)
Now, we know that the transpose of matrix A is
![](data:image/png;base64,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)
So, from equation 1 & 2 we get
A = A’, hence we can say that Matrix A is a symmetric matrix.
Hence proved.
Question 14.Show that the matrix
is a skew symmetric matrix.
Answer:A matrix is said to be skew symmetric if the transpose of the matrix is equal to the negative of the matrix. This means that A’ = -A.
![](data:image/png;base64,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)
Now, we know that the transpose of matrix A is
![](data:image/png;base64,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)
Now if we carefully look at the equation 2 we can rewrite it as
![](data:image/png;base64,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)
So, A’ = (-1) × A (from equation 1)
⇒ A’ = -A, hence we can say that Matrix A is a skew symmetric matrix.
Hence proved
Question 15.For the matrix
, verify that
(A + A’) is a symmetric matrix
Answer:![](data:image/png;base64,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)
(A + A’) is a symmetric matrix.
![](data:image/png;base64,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)
On adding them we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOEAAABACAYAAADs8xAxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAeESURBVHja7Z2/bxs3FMcpe2ycKYtTU1OLAJliMdozBPK5cbwJjiRoMqDh4AqxbEBDkMrePLjdgwxxPKUZujaI3b1N/4CkaLy1Q23/CYnTI3WSzvKdRd4dKZ71HR4CCI7E+5Kfe4+/3iPb29sEBoONzyACDAYIYTBACEsiICFTUZbBZ8ld8jw56KlHS4CUwDqd+iwj5MgT9fNFo8dOszOfpedxHdry2v4p5HnOaGltw5CeH6+CnipaAqaEg6ZACgf1Tmf2Kj/nWolumILwqusZpiVgwqABhIBw8iAMzBdyJtsbd55lM4Tj0jJ87vd6ChBaDiHvMNcts1KJ/UAp+cvEwO5Zu12/6c9fvwyMnsjMs2yEcJxa9n6/0Shfrzislc+T92K+xyp7gNByCLfq7AHJ5997nfaBQ2By4PDOv7BIwKovsuoJx6klB3CtUqxQkjshlP3u1JrfNXZ2riEczVA4ulVjSyYHjvCC+YXn5UbjepywzeZw1LSWwgOX6Kb3z2fquC3MCQGhlFULZF/8HmVvSivukuqeGyAMtLHCHnoAnsUBEBBOKIRbG84dSshxb49KzAVp4c9yu/01IIw5r6bOgR9VTKseJACEE+oJubmu+03NKT7mQOYEjOyo3m7fBITK8+ovRae27tzNP+tvxtPCu7LbviUT6gPCCYawZzs7jWuLlLwVHrFQ3QeE8ro5c+SQryh7ED7uLcSs1xxPTnLifX56331SAISAULrd3eNhnjeUaDsgDBxRnHMOh1dCmw5tiejCe6mN8oaAEBCeGzj87Z3VfULjEDadeeHxWPVlkpcaIASEQ4Oq8A6eUDF6CPF2g1AVEAJClbbzpXbqHMhsMgNCHzRKDggpXFjQ6kMooScgtA/CTd1t7J1tHP6sxuiu7NWgjECoXUt+SofP/eYrT5eD3vCJe7/AT8+w2tYSFmYyACEH4mllfrm7Ovlo399ryulqI4ctzxae+0vo0+12Y4Z/Vqw1V2S/w1YITWvJrcrIixyhJ/6hh+m2W77FKPlD5tzoxEIY5gnGNWiGNs77Fvf0hdQCTK24EvzN/F3nWaXZ+VblO6w8O+ppmSfkv+GLszq17OvhFFcHv02Pi05zNYmWI//Ta8krGqZMtT3VAv1JdhXQVDiaNcN9Qr1aSgxish+2NzIuU2mPv6x8Sgj7qKNjASH01A7h4A6a3GkA3abanh6E1Gm2MGgAoQ3G92eVIFwTVzZErJ3K6pM/P8slGAxK7emunOnxgpMEodAxZJM67X6eBD3DtIz8497eR7FeXxWblCmEpEneqHHaI35PkxcEhOn3MyAcMrH3QRdf8YFeY2QvjZA0CYRx2tN0nLLODgWEgFArhFVGXva8SG9DVPbUvQ4IdbRnRILWkclvASEg1AYhXwAJihFYZTxKIlDcztHVnu4h5rBEsxctLKwFhIBQG4RiAaR7AkF4AZ5VqjRHfuPeh1U6D+N6Gf69fEFFNcV6Gu1BOKoHwrB+dkU/u5vBm+cy/QwI+16nMbMwlzuM8AjSIWCElznz7dznrL71INoLptMeQKgHwoh078r9DAiDQkScrk8jBIwTpuhsD+aECEethJAvgISdBh9c5YgfAsbpHJ3twZwQEFoHYfdKRvRFz8Gq5KN9E/krdbcH4SggtArC/mIHc37h6dzCQjcPCtYNAbn3WauqHuxW6RwT7QGEgNAaCP2r+39fVgtOpB2PmUY9GP7JnGAx1R5AqAdCV/Sz+pUihKMwQJgShNATEAJCQAgIAaGcqda2i2ujtl9GPiffJrIUQtO17ZNqCQhTHzT3bt8gN/691+ncVoUvUNvug875bD9p7bm6hEEbfdXre/bVj4Quv7FNzyQ1F23SEjAZhrBb245VRSJZUdtufVFn1oJuFaHI/c8zwmp7o77DVgj92hDnT+tIJlyKHUpq0BIwGRw0g9p25MxEQiJuIrWhB/rw593DDvRnGa9hI4RJay7apCVgMgjhoLZds2WmffXZSiV8YATvZ2YRQn6SKknNxXhabt3RoSVgMvnm9mvbNdrtGVMLCZeFcrIvA9sgDKQ7jF1z0SYtAVPyQfOPDIR+fpzI2nYm260SPgUGzq826cktrOaiaRBjaLk7rCVgShiiyCypB2rbnSapbZeWqYRP/Te9xdnWejUXc2O42paGloDJwKAZVdvO9L1I1QRYWUh5eK48mWQFYlu0BEwmIOzdfYyubXeUNFWHrvApKxAOXmj69gl1aQmYTHrCEG+nUttuHOFTliBUqblok5aAydScUFxAZolq240jfMoUhCILw+JbUyUb0tISMBkaNCNr243IvzKu8MlGCNOouZiGlouUvkpDS8Bk8M3Na9uR0Np2Zu5BCuBZbTfWG98iCNOouWiTloDJcPjUdIqrg3qBarXtkhp/CcTxuLZByGsuBmsTxqm5aJOWgGmMc5isGKoy6dUSMGHQAEJACAgBISCEYdAAQkAICAEhIIQlGDT85H5I8ZOprD1PVFmAboEXcxBeBT1VtARMCQeNnx/108V0BxdzpdpuEcVd/DIA+jMBXCU9VbQETDDYuL0mRIDBACEMNtH2P5W5aF2IVopLAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Explanation: Now to show that the matrix obtained i.e. (A + A’) is symmetric we need to calculate its transpose and prove that the matrix (A + A’) and its transpose are equal. This means that (A + A’) = (A + A’)’.
. → eqn 2
So, from equation 1 & 2 we get,
(A + A’) = (A + A’)’, hence we can say that (A + A’) is a symmetric matrix.
Ans. Hence proved.
Question 16.For the matrix
, verify that
(A – A’) is a skew symmetric matrix
Answer:(A – A’) is a skew symmetric matrix.
.
subtracting A’ from A, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Explanation: Now to show that the matrix obtained i.e. (A + A’) is skew symmetric we need to calculate its transpose and prove that the matrix (A + A’) is equal to the negative of its transpose are equal. This means that (A + A’) = -(A + A’)’.
![](data:image/png;base64,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)
We can rewrite above equation as
![](data:image/png;base64,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)
Also, (A – A’)’ = (-1) × (A – A’) (from equation 1)
(A – A’)’ = -(A – A’), hence we can say that Matrix A is a skew symmetric matrix.
Hence proved.
Question 17.Find
and
, when ![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
(i) ![](data:image/png;base64,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)
![](data:image/png;base64,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)
On adding A and A’, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVUAAABgCAYAAACtzOGEAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAA/XSURBVHja7Z3Bb9tGFsZHm+Mm6SU9NKvRbZtFsIeajO45pBJdJwH2INg04VMKHQgtATsudDBSOTcDTbvnNodNfSqKxd7a7jrudZvsnxDbt+6hUv6ExFk+kiNRFEWRlEgO5e/wAbGLWiO+7/04782Qwx4/fswgCIKgxQgXAYIgCFCFIAgCVC/GBWXsd9OE6yNVnCoRsarAK1Bav+CCLVC93tYHKmNn9kV+Oyne16zeCq6THDI1vmPH5U1IrM55o/MwJ6+cwivlUEfj2xF+2fWDFRdswYmiMOX5Vq/3Aa5HobOJ1DPNToM/zAuq8Er5/WLZN2fyC6BagkQJlhu4vtFqt1tXdU3d4Yy9Vo39u8sIVXhisX5ZF37Re/cA1SWHKgXIXG/cVTl76ZQcXH15x9xT8+r1lU0HB+3LTc6ObOKc29fo3TJClWLf0RtrtidewBPz+0Xj7N8Vt3x/B6heAKj2dPUeY3ygGdur9PO2oa3ad9TBPMG/CCIgLitUPU/0NcP6xOeJ/or+6D5inx6I1BMFVJccqt3u1nVa7OKatTNpAPVsq9u9jmt9saBKnlDIE4FxuZ5Q4AlAFVCdkdS7ziw1sPrbs7QV6v/QSqNM37mo7UizoBroP1aS/A3ZoOp+r6meGEjoiYosnogL1e9TjhdQlRyqTq+nyo5ZVTtuHxxcDv5tZ7uWuvmtLN+Xmv1NtfbUtxVoUNU294tKIgHVut7RDY1/TuWxs+VF2fiq1e5eLSNUY3jilDwhC7jIEw219s2YJ5q6TZp30oF1BNXOJvmlxthvFccv63+L6xdAVXKoDsEZkkBRyVWEhuNh6plu9f5IvyNjztv4XwRUOV/9jsZEY6QxOQtYyuZhHPDIBtXYnuh2r0jhCc6ey+SJOFAN+OWR65eNwzg3AkBVdqh6JX7YbNQPMRn2NooVd8abR5TsZCp3McWGWg5QmlX+TyY6f33H3FNKB9WyesKGvM8T50V5Im757/8Oq/QdYvoFUAVUM+mfUclnGfUN6u/JBlVf8sSaLUkK1UERnpjxaObUXnrAE/2sPZF2nNMWqpIsYAGqZSn/QxIoqgwsEqgdva5T4nBVf2I11E9lhOq+od6NOy5Jy//QvmnWnoh4lHdCXDN3IjyR6Uw14hHS0HGK6zgNnp5fzoOwBFRLCNVhqRq5UGU8k+X7bir8K/+qdBJ4FQHVOFutpFyoEp4I9E2l9ITKvwzxRGnKf/+YMVNdAqiOAj25H3XPvKPM+wjmQr9rSKtCVqhO26ZWBqiOPKFM88RARk8IyJQRqmILG3qqSwLV/YfaR2LvoQiUU1I5YFBPZemn+gA63CPpW6jaLWKLzxCqviTpdttXmtR35KtHcUpkGaHqeaI/xRMnknnCBqj5mRinb6FqV7b9qgKq9FSaGJvjF6oMvMU2QHUJoDrqZfG+92z3JdGjUrf212T5riLRaVbd6nb/sGe2PrxVq/1AUKPtS5alr4htNbkliVFfd3pn1eZxy+zeoMUSQ+O9JOCR9THVqZ6QZJY67gnlrGVZVeEJsaXNNFtK3p6IjLXPL39pP/zTyC/KSdyn1ADVkkBVAMJbOX3LuPpCJjOODFV/4I6R9+ua9cDr8Z04PxvWehFjomfihy8doYUJVX+SZBFH5rdUkSdog7r3QhUpPdGxPeGOceQJhbFXnic2ZBuv55dfxvySYM8voFoiqEIFQQHvU4USthAAVSQKBKhCgCqgCgGqEKAKqEKlTRJ4BQJUAVVoAXK2BeXwJjB4Zbn8AqgiUSBAFQJUAVUIUIUAVUAVAlThFUAVQqIAqvAKBKgCqhCgCgGqgCoEqOK6A6oQEgVQhVcgQBWJAgGqEKAKqEKAKgSoAqq4HoAqvAKoQkgUJAmgCgGq0RLH1wKqEKAKSQHV752zsr//nSxfIul4vBM/X8c59A2JMt+NSwhQvTheWabY5wbVTYUdynTWfJLxDE93zOnAvIsIVTKTud64q3L20jvm46V3jlIFUF1urziHD+qNteHxNSWPfS5Q7Xa3rjvnitszvTjHtWatpOMRUOWatYPyP6Pv7JyWyQeasb1KP9OZP3QaLJ1QCagut1e82Pc1w/rEF/u+/3RSQDUg9whc9jZ4FPGcpULqi510PO6Rufkd63zRoCpucsGblnv0r3oW90TKMkM1rafL7hWKvUKxD7zQ24294pywC6gGdHDQvqxV2XF9a+uBnTini2gBzHNURZrxOJ+X0yw1r0Tx9688FTYjcG9y9iw10K8eVggLuBEv6jplBdW0nk7qFZniLr53ROwHRcY+L68khuqeeUfhfPU7ApehsmeLaAHMA9U047E0rZXnTCBrqNK55E219tSbrb8lU1e1zf0iEkzc5MJubt5R1ad5lNGhsyizdeMW5z+KY4dravNpq929WlaoUtwbau2bsbg39cdFgXUs9oHjnP2xL0MLwPFKrfbDyCsfx/ZKYqhuquxbMctzy2j2jimbh0VBNYvxBFcuZ6iyyERJbWS7rBbnvRsa/5yug2rs383bjF7ynIVBNQq4Wcu5+bLKoG5Y645XHmof0cwp7lhkg2pk3PXevSJANDP2nD0PA65s8rzSrxvmhs8rfXvsP8cde2youv2SUcB9q+hn8wAjrQGzGo/b/xF3/2jFaSP0erdvXmPXfr3d693MAqpNzo4Ybx6RkSmIj/SV+15/+WHuiSViEAIgPwjyrBSGn6saz8ZvyPwLeywnccayf/v9bfb+7V+yh2o8r4TF3V0gKibu/hJ/auwJqpL31P1e8QPR9YryKu7YPb/8ZyZUnV6ZsnEoZmlO+VFlPye5O4bNAunvUq8l6SxwEeORrfxPO0umf9P3t4z6hmNsQHViTPPM3KNmqmExMx1Pm+TpS0k8naKn6o97X2qoitjnCNWk+SS+w7z8iDVT7XbbV5rVyvGUGVvsknvKLPDc09jv1a39temz1MWMRzaomu71eRNvlmzuCON09LpOScVV/YnVUD8tDKoRfdOo8jBLibbQPIkSBdUpMQv39AywJ70ByxL3WX3T4X/LOfYdjW8nySfPK+e5QNUpLbj2fGLxYQEld5pSKcvxlKmn6pQm7tNhwxVXAZEikmusdxa+UHUWLMPzguo8K88yLlTJFPdZfdMRcMfLatm0v6WuEVTtKuOzecYZC6q0IBR2lx31StLPBNIYMMvxLL6nmh1Uw8rtopNr2n5UdwFgvjJ8rrLUHlNwn2Tc/aSyQVXGuI9ir0zEflhWF7B4mgiGYgHT2VPbS+WVWFB1k0H577RAj1bdnf5mYronNWDW4ymy/F/ELMy3YLFbxPcXxvR/vlOqOvtX4y0MZTKD8jzRarffE71OulZRbSZZoSrKVP81Ljruvtj3/TO9ImOfxiurtAAY4pW9mF6ZCdXh4o+q/dP+gKthpbINOdUtuWl22NlM+qKVJAbMYzylgurwzurOwvbM1ofe/jqnr2xZ+orYcpOn3D4j73vPfF8Svb+4psyorBO9zneu+CCshVQKqIptPjSjsqyqiHuFvp8dd9NsKUXEXcS+EhZ7yWepwxi4XnkT6pV5t1R5fZAT3+bifvBJCZ9ZR1I3/560ZIhbRucxnjJB1b1+9QdugvF+XbMejK6T/bO3L7OQUtCor3sr0vRSjRdFJbkfRMOXfNCG7lva13HHlBSqpuNpcydLr3TsuNcY+80fd4WxV0XHXcTeHRvFXik89im98svIK42vk3wHvKS65FCFckgyvFAFSuEXQBWJAgGqEKAKqEKAKgSoAqoQoAqvAKoQEgVQhVcgQBVQhQBVCFAFVCFAFdcdUIWQKIAqvAIBqkgUCFCFAFVAFQJUIUAVUMX1AFThFUAVQqIgSQBVCFAFVCFAFQJUAVUIUIVXAFUIiQKowisQoAqoQoAqBKgCqpPyHVRYwfW+OFBF3AFVQDUDqNLxMrqm7hRxoF7ZFDwlt8xQRdzl8AWgumRQpUPKmpwdVbyzdJBc0YnT0RtrKmcvveNdXnpnKMWa5TmH6kkCVV/c3yLuC/PFizS+AFQLg+rtm9fYtf/d7vVuZvH36ZA5JFcMKDI+0AzrE/p529BW6cDEFf3R/Tj//1/V33/J+P1/yeQVxH1hvugHfNEnX8wDVtsvT8gvgCqgupTqdreu21OPs+Dppu4Z9erEGfWA6sXxhcLYKfnCDz/XF4pzOjGgCqjeDfSHKjJeD3//alE9rBnXZ9eZpQZO3u1Z2gr1JO2k2i01VPXePdninneM0+fNVF8MyBdpryWguiRQresd3dD4596x0Odc2fiq1e5elWp2YLZu3OL8x+HRv2rzaavdzmyM1H/UquyYVbXj9sHB5fG4OEd5n8bplcoMVX/cKxLEfTLGHz+VzYdjvuh2r4T6Qtk8BFQvOFQ5X/2OziYnw1CSOQtYcxhj0doz7yicVQbiTHo6W51mBGHAW9y1dxLkLOwzooBbJqjKFHdfjDcmYhyAV7E56YEzZFyOLzh7Ps+YAdUlKv8njMH462B5U+jMQDWe+X+/qfIvGFNPstrDK0r8sNnocEzUV53x+WXpqfrjfsfcU+SJsXIi0wMNosQPrtCPX8N4/XZAVWKoBvfLzVAlKrmifr8oJRmvO4PJf0/lRYMqyV1ocXutRVzrMiyczYSq8AWgWm6omm4yOH2oWeKauTMruZy9clQeBla9F6Uk463d+vM/Ckn0iL5pVGugzFDNOu7T5H3ued4xnssXIW2SqNYAoHqBy39/cskwc/Al+m6enzvWHwtfqDoLlqtLAtXzvOMuPneeVfNCfDFtocr2BRaqANVg+R+6lajQcssuqYL7/7w2QmZJOG0/apKWRJmgOm2rUOZQFYtSzh7PXq4xTu+Lyf2owqvz3JQA1WWBqq/s6nbbV5rUF+KrR1mtrKeaGdjjZMrGYavdfs/+9yXqt9JTLerW/lrWye6fQTmPJzo3nXiLZDJCVfRO/U//jOLezD3uFONVzo6KiPEcvujzhvlZqC9S9lMB1SWAqmXU153eZbV53DK7N+hlG4bGe1muqqcy8Za65vVYnXcVuOIDxrXnWQPA7f/yvvdc96WOXtcpoeImupRQlTDuXozfFBHjVBMSjW9XyBft7i3yhWn7osbYb6rxaK7WCaBacqiS6Jnl4UshaCFL1Z/IaGKaHfjHWbulfU17LHMp92wIUcJ4L854keRzZYSqrHGfjHEjtxinAuuYL5QXixgroLoEUIWylaxQhaT1C6CaQ6L8ikQpPVR/glegBFD9CVDNLFFwREbZRYuBeez7hFeWQ7SIGHz7FS4MEgUCVCFAFVCFAFUIUAVUIUAVXgFUoeSJQk9uiM3Qsr/A96Jq2ktizAbfzROqLcuqwivl9wugmmGiqIydhL98hPdleIwUchX1khj/S2wy9soreKUkFYzGt6f6pWECqhAEQZnNanERIAiCAFUIgiAp9X9kl81w9GjG+AAAAABJRU5ErkJggg==)
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)
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)
Therefore, (A + A’) = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(ii)![](data:image/png;base64,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)
![](data:image/png;base64,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)
On adding A and A’, we get,
![](data:image/png;base64,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)
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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQgAAABgCAYAAADhJux8AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAxkSURBVHja7Z2/b9xGFse5bi92lRRJNOpiA6kijrd3YazGluJuIXEJVQ62WOiIsyRgC8FauTNg2X2SJlGVu+LK3CH2dYeDpPwDB8gur9HqX4h8fEMOlxoNKe4PLmdX3+JbyFrbb95yPvPem+E858WLFw4EQZBJcAIEQQAEBEEAxM3+Mh3nVpbgHyh+Rmo5z0kNgJhT9Xobn3PH+RB+yX9cFTsTQW8JfoI6gm2ZnxHngjU2twGIOQaE67hvN3q9z+GPGxsV1Eb9NzYbbBuAACAmFprOQfpVmwe7u932bU/wLeY459zfXwUgoKkDgh7KTa+xwplzLMNRxo8fdnb5rEyy3U7zrsuck9Dej1LMPWmsdVZtt99g9+9pu1++bH+yzJzfQnJf0O8BCKgSQPQ8/q3jsL7wg8f08zNfPApXrD73et/aD4eHLtmq59tyQllsf57dS97zJ/rkBiCgSgDR7W58QcVP/eEJZLGLf9jodr+w1Se0wgrmvGXC32u/fPmJ9FPgfRXS7bdosoX2W1izybO7puxO+R2AgCoDRPTwsXN9F6QXiCXKe00PlTU+CW2si+CpcQIuOO/0cRm2AGvW2R2CQ7c7AUQYEWljqAEQUGmASCbSgninVrL0/ye3VXnr51n0V4s7P4cr8XvlLyr2LfPFH1Jbw/0F0dq3rU6h250GRN3b9HzB9kJwn8ltS3f9Tbv78jYAAZUCiAQCBkDkwcN2JbZz/6fLEQX/4AW9r+jPaKLZVqfQ7dYBwdijX8h++hzZL9MRt3VYBHIABAAxUqhLaYQpSkhPqlk7exEVAPmxspvGIusSTLwl2NGEigqz4aSzKIXS7c6rQaTTkYedXReAACAAiAKiyd/i7LWpoEe/a7ebdwK/vh7vIFgDCLLN5+zAZHdWkTIqJBeLggAIAGL0FMMAiLz0w2Y4yIkggi3j77y6R/k74/5B0ODf2QKIPLvzALHv89WiYwAgAIjRcl4KU3OLlJfzYZtFE8a0O0BqueyNPOsR7w4MM7mqtLsAIC6KbH8CEADESMo676DSj3H23qe5AlP+zrm/Z/qdKZWyARDX2X0dIOIt6n6RF/UACABinDpEP3x4drSQd0ffbrMVDlRwZG7rjf5eQ7fTvMe5eEU1hxgGyRhTRcqdKrY6i9itorr0OQj19+W2LdWI4sIrAAFAlHbUOooiWD9+/+KWytVtjx5SdYXz5H0GTTQpFAQpUmp2u1/SqcX7jP0qP+O2Djudpqu2P22yO/l+/PqaPLuxsPyu2eneo0KrL1gvHM9p0ecBgAAgxoNE+BDGB3DoZa2jaU6YcWzOuPvgyj0Zgag/jcbHzijfj2ssp/JnP1i31W4lej+GM+dIfYZx72CY4jEAAUBAEAABQAAQEAABARAQAAEBENA0RIVoAAKAgCCj5LkPw4lZOAeAgCAAAoCAIAACgAAgIAACAiAgAAICICAAAgIgIAACAiAgAAICICAAAgIgIAACAiAgAAICICAAAoAoQ6rdGQABARDlTrLaLNoS34h8XuRy0FkGhH43om1t6UYFu5LFdpbud+sBIXsOWtJjYRhbkhuRK77AtWxA0EWpLnNOknsRmXvSWOusziokonsfGyucOcfxNXrH8Z2bVo1nt9O8q/n99zL8bjUgVIv5om3CbLIl6XKd0dRkHgARtXyTnabSdyNe2Na7cih/yVurWV/4wWP6me50pDHaNJ48vy95z59YD4hJ5d7RFerRwNNXj1ehYW2J+idUf/17WYBQzXOY8PdUREW3PsteltJPs9ebUy0C+gUpWT1AqlCe32vK7xO0sxRA7G/wFTk5xjBU9Xisb2w8rbqV2yi25LVEmwdAUIRk6uo06M15ufai5/U2piBZDWWSaLDiRepav8dNeU01L83/tYkCwlAMuVby/n4m/tbudm+PHEYx8VeaiD53fqoyzRjFlkCIpg0raBW7GLJWk4qeZLMWvvhD6nr2/oJo7dsEiQRs2e0E35smik3S/Z6uV8T9PKT/F/nyj812+85EABHT8yz/Pv4cMX7c7OzeHWWwagVW7c6q+oLKsMWwouapNiuASCZa3Jsz3e1b9czwBdsztYSrGqRZ0WEePKwDnNYTVdUrVA+P/W3xjaxf0FgKLN6lFykp3Vh0Fv87TCWYcsH0Q53aEZh6XluWLXEL9kKQHSdV6fUefP2p8+n/HvR6X0+vgMaPlW/owZV1ibjVm2odZ0sDXD2NME2GNORsravofs+DhsfZQdHuWvsPPnvmfPbgP/rcnajxUU3CuaCwsgi1ZEHQbR2qFZRahoUD/VfR6vgkV+dxbZmVFGMSPqM/b3H2mm/sr5h+R76jfpdxBR6AmJDIt3446fWIbBKNlKeyzTlo6np9y3HKV6Mvw7iaFgrtJ7U6T8KWWQHEuD6Lv+PtzN/FfTsZ9w+CBv/OOkDk1Bny0g8b4JDld5UOj7OQlQ4IFVJSmlEX/l+uc7AMPw2dh6tIM8q0Zd5qEPQgmarrsoYTnShNdgfUg2sTIJJdgOwi5Qc9VLdBeX6P/Xwxzu5LabsYSrJVOWP/LnrcmAqCpihjsI0zvdC+TFumV4MoFxAEL8p/Off3jIuDIXS3ERCD7+TqOYJJhOrT9nvK7qQ7uWGBqo0NiDF2MS6k2OXCyfWFFvck6/PJDkJUE6iV+QXYZMs0UoxxokPmtt7oCwMdweZcvKKag364zMYiZXpCka3qOx2kx9UfehvG7xQFpRcyx10/pK3N9MJdBHilnaSMzkE8+qVozpYU/7j4u2mPNiYmj0J752Pd2/TKmpg22WIrIFJ1hfPkfQBNBAB9FaNTf/GevIRrEHhLavvTniiC9eNdt1uqdmJL9FDU78kEjzYI/rj8GdY3pc5TAwQZxdj9X4s+lHGOd5o6SHOmpySpgQ5UQk5oky02AyKMDNbyI8iB3wJRfxpFoeyM8uWBj8Of/WDNNp/R2JKomfEjqwA2hN+T53VbfBOuZkfJQan74vuiY8J9EHMu3AcBjbXYAxAABAQBEAAEAAEBEBAAAQEQEAABARAQAAEBEBAAAQEQEAABARAQAAEBEBAAAQEQEAABARAABAABQQAEAAFBAAQEQEAABARAQAAEBEBAAAQ0l4BIXdBbw3cCQEA3CBDX3cBNrQM8wbdsu/R13HFZbPel29IBCKgyQNAFqS5zTpK7Dpl70ljrrKrJlHTViruizwogqJdl3rhshsOm11jhzDlWbS+H6WgHQNxIQJTTek/1fDTcZH6lFUDUNXs2ADHMuKz7ruVt4awv/OAx/fzMF49oLEve8yej/pt/5n967bAn/yy19R40X4BQ16kz4e+pm5HptupBtHC5odCsAGLYcdkk6iFLzX30VgJZvT4ACKg0QNBV9qZuToMeluw8fbNyGhBajmxVyD7suDLy/krGFPl40L0sPSbV6wOAgKaWYmSJupHpTWYUIKh/iC/YXnyl/AU1finSzNkGmcal6hVxn4/oSnm+/KOpl0rpkQ8BLLtt4PtR+8gCEADE5B9UrUeIAgQ1UqJ+DPQ5AoUM2y3vTJY3LlWvqPvBenq1nnaT37zmwpfgMQKMAQgAYmKKJszVVoumGsSgLZw5bLdJpnFlQcPn7CCMNE6nWasw9T+9mh6NVj8BIAAIUx49dLdx+rMWZ6/5xv6KOT++WqQsu3g5iQ7qybg0G21q6JtELgAEVBYgxuk2Hje43c7qQp4FgrK7fI/bQT1vXMp2G7Y+8+oMeekHAAFNJcWgyWKq/hcFhK3bn3njSsFtp2o7k3TNUGdIADFi71gAAoAYK4Sn/Jxzf8/0u2Iphn01iCLjSnct188YxKlNbZo2Z513UHaOCmEAAoAYeRLRyT3aqtTzeTqCzbl4pULaBBCpcJze0ZC5ccE29NOEQ5FxDYqszkfHXT+krU31Ofr7046KqIO3Ou+gJnOcIu2YtmcBCKg0QERn/uuerJyr9xU0pesKScv6MARudrr32u3mHV+w3rSr/ZMe1/4GX4lrGKnPsH5V0OvIKIL14/cvbsVjORsHVgAEADF8OKsmfKbYmZ420HsBnDlHSVGQ+wc2RQ6jjotC+PS4Fu+L7+msR5VjiA+iyZe1xrUFgAAgIAiAACAACGgkQByEgPgHADG3gMCVc9DokudADOdU4BwAAoIACAACggAIAAKAgAAIKAsQzW73S9NLSvARJCd8xottnQbbASDmHBDccU6L7utDN1OdnBfbmOhsARAQBBWPOOAECIIACAiChtb/AVxyiuiaviLHAAAAAElFTkSuQmCC)
We can rewrite the above equation as’
![](data:image/png;base64,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)
Therefore, (A - A’) = 2(A) (from equation 1)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 18.Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
![](data:image/png;base64,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)
Answer:As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, on adding A and A’ we will get,
![](data:image/png;base64,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)
.
![](data:image/png;base64,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)
.
.
.
![](data:image/png;base64,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)
on subtracting A’ from A we will get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOcAAAA1CAYAAAC6AvwEAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAh+SURBVHja7VxNb9NIGJ6UK3AHOrkhrltP+g+Q64qPW9QmUS7bJYeoipYWyQcEKbdKsJyXvVB6ofSwR1YqcF/gB+wu0D9A+QmFrt8Zjz22x/Y4cROneg+PqiaOM3nmfd6vmTF59OgRQSAQ1QOSgECgOBEIBIoTgUBxIhAIFCdi+sZCSM3DHPxFPlCciIqIst9vMttmDykl/1F7fRN5QXEiKoCtLrtB6vV/6nXyryfUExQnihNRNZF22E0UJ4oTgeJEcSIJCBQnihOB4kSgOBEoThQnAsWJQHEiUJwoTgTiVMSJO4uKcYFEIU5dnGCIw4Gz4HTuLiOPAnc7zvL1/n0rS6RIFGIUcd4r8rlhi91adAZryGEUbUYfs9bwVppAI97N39TsY38uwxPOmV5bJewnxi0wyVRmVjmGMTxoLdwGcRJrddf7e84kNbvfv25RurzX294+bz5H1benIuNU5jHC1/Z27/wypXteBGWZ4uT7Jwn57kNOQIJ8SE8oIV/9637wv3TpwJT8aQGIWKLkIPh9cuyEfe4Oh5cmUUv4HB9HOT5JXL+16fxUJY6V8RwrNvKdOv2NrM+5bu/C0nztLetu3TCPJuQFmXfe9lz3QrWjntk4h8PuJYuSD4TQbxAldTZRm1/S3idxs0FnccWbiCOZvqQZG6Q4lDb+KtOwJwFwLoyQQ0KtD2WPHaIEswe/5F0X5bh/5jhW01luwIZOxXW7l/nceIYM9VhVf1eRcYI4vWs/cYem4cJ//4tOuNobrtt00/fYJwutB7fTjJxS51XVI2YKWYdFjMY0Yq5aZNe0WRLnODVLmUGOZabizJO3hHV2TD/jcXJPZhVF69pJwp+7QuMEQTfqzjPdXHYY2dFF4dQvbzQavwvj4d6BFTGcZG0VNbx9Tc2XVY/tl1gfjirOrDoV/u8Lw5KR8FxePTIux1UHrzUJ+cY6WzeLiHmx212DSFLV1JaPk5I3yjjf5Y0TbBvmO60MEI22ZBRONRwgdeDQjRo3HnbYdN0rJobT6zUvthy2EdZM9Iha7afNXu9iWASTg5ofNWTNJ2ve4HXf48rrhRGPXx+OIk74TU6j/iyot7yUuNl3rylF/ctELcvaz/PEOSrHswARXeg3ZzBcMBaz3zjikaSiqe0o41x3FtcWO4OVrFILHFk8Cmcajgy53Hjm7YiH0BmOaAB4qQx13sjXob4isc+DQet+GAgHvFL8dbivzewHrcHw6qQjJ09HKH0t1+iGg9ZV7jE9MXVd93LU+5mvAaoctxl5Ljlu9tyLRcWpyVSyUJuEEfOGCXBk6EzheuoMNlQuidXerWIjKDbOHzDOcXj1xXnkOfQXhcSpRK5IB1dnOKJmSHpLiA7x/DzNW/j5fMTIobEgCZm0OPlkiMZYYOBhChuOfRxxJjk+qRURZ1/wG+mkpiGvw1pqvWkoTnCAFrHeyGsDY405wGo0gthBcpzWWONMs8lcw1E+/Fnt4MYNJ5gQXUfKF6L6nszd49cLjyu8Ufga/c00PSpTnKGRyWWXGJTUdRxxJjnuazk+q0037tTDtdM5KCOWBO8nui7mtLIHHjjGHGcGX4lszMhwol5CdBfjhpM1IcF7MU8qImqYwop8vvGaUfJeXiuiyvKTsgy0iOHIa00EN644+T3EemLAseBjOuIsYOiJJpnOGadHI7EWmhLpjVPb084eTMY5qujDIBDNOo0NR4qp5ncX7ZZ9x1ScSprzRRVnMIn+BPDv7W7dkGkweCN+TUkprak4g9RdXptiIOpklCFOaWRpHE8yapgauopRxMnXQpUeRTIYVCO1zR/n6KltKeKMNIjAW+jSVE33KjDy2JpX+Bl22LzftFh96Q+4XxDmPVGs2+xOWSmtqTjXbXsFvlNNN+KdVF5/sM7DssUZaRDFOJ6VmrOIOKGM0fEQ2sZ4KWOZjaDTGmdp4ow0L2Lky8ga31kk1rxqR7ptXEGUZPSVGiFlN5c2Gnu6psL+iPVDnjhdt3lFOolwHN5vpdZHWD5Rm0IqR/FN4XnjGpXjGao5v+Q1hIRdpO/UCru2q7tkisfNxDjZ+5xx/hh1nGkNNG165O90uZdm/EHzImY46poka623ZdHMPUvKTpHAy8ZS3mBiNJ9TPQ2z1++YkgCC9ohmsv6FaKimfG6/ec2eJ+8ineKwUeOvy8q1zOiaa1AvehMEv7njOD9nbdAQHMsNC+YczwJMurVBM4XZf8o18DhPMFfCNiAqgT1NfkN8fJzxuSpjnEbdWj+KRdKgtE3LoiG0/FJXX3YYfRLeg35ddDq/phlYsCE9tmgvjVP3/fBeuIk9Kuosg1E2vmci/p2wtqlsNPheb+i3YbXE7z4mlP2dtiarS0HTOOaC13A8C5CZj64kURzesbSR+HWxQwJ+d7y1M4UM4NNpjzMs+wzWOWfLCNhwVjeGn2XITnyZ/YKzCmXNf/NMiFOuA+Ih3kob3FEVmjmV5wo6wYQexR3ZTP4Ynjp7NR0+9mLiDtF4WSboYlZwC17VwDfeaJZpkByEEaBZJg4N+00xan20V/o38w6X1yq2Ba9yvMoD6ZrOPRKEyIV//OtIbXrUcs77yugJ3XtMbXNSWrqsfcoFEoTILSEgPaVO52GwG4x3r+Wxv+y1TL/jvFf1x45MK2rCM4TSmmZIEiITIC5d0y1tV0vaPWA3FT63NlK/1wbO4lrW85WQKMR4jQzDdWYQKDbwQsBza/OcGhKFGD3dLfiMIETB6IokIEZB3n5TBIoTMaV6CQ7AF3keLQLFiZiAMLOeJIdAcSKmBDgthFsmUZyIikVM/kR75YC5+h5yhOJETEmY/AmI1upT+XAr9QwsY87jWTzWhuJEzH6N2WJteZhYhzKf8YRAcSIM4T8UPOP5RMkDyAgUJwKB4kQgEChOBALh439QWPtd5DYK9QAAAABJRU5ErkJggg==)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, Add M and N, we get,
.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAU0AAABACAYAAACeGSWuAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAvhSURBVHja7Z29cxvHGcYXVBlbadxY5qKK7BlX4S3YJRkXCni0ZXUYEcSwcSKM54bBmBQ9KGQRVMdJKNeSXJhSI4tFutga0eotuU/iSOySwqT+BJPK7ddh724P94EjAYJP8Uxk5nC7d7f7u/dj9z1y69YtAkEQBGUTbgIEQRCgCUEQBGhC0GROQkKmkjSh11sZcM0VQBOCoET1ektvM0Je+rA4jIvuu53ezKRd87JLV/zr+8VyzUe0vnwd0IQgaCA0HeLsLvV6b5/1e7Fcp9cBTQiCAE1AE4KgUUFz3GKeZcZhAU0IgkqF5g2v8S6j5IcgHkid5w2v+96o+t7tLl1wCNnz+/K6L3pQNA4LaEIQVBo0b3iXnFrN/Wt7c/MN8btO86IEqLO31O1eGJU7HUvosOZ20Qw4oAlBUCnQ3Nxsv+Ey9wsNzOC3HXeGEvKKtTYuj8LKZNU/ftVot8+XtUwI0IQgqFT33GZ9UlI5GAU0Fxm5z91xStnj+lXvchnxVUATgqBjgyaHVJODy3eHT7rPG9fd31YJ+VmvrRSxTDrzY6PbfQfQhCBorKC548PS8xrMNzGfE1b/e6PdPT+qvnue95uWO/sZB2hFwNPZa3R7hcEJaEIQVCo0eWxzjpInRvLlNXUWvmx3N98c5TXofglwOosPkAiCIGis3HMOqU5r9iolZF+As768Nuo922o76AtCWOFsPqAJQdCxQFOLxxZ9cB4IUI3BjqKOS1exThOCoLGFJleLke1xgaZcAsWeFe0LoAlB0LFDUywwp+5udA3nKLTRYpcJnX9StC+AJgRBpUHTtq+7222/6VKyS11v9UShoephRv/WYnRrmFJ2gCYEQaVAU2yZ5DU3qfNjw/P8f5Jz7XbjPIcUZc2tk+4zb7fK5r7i+955Xzi8+d9mW52rw5wX0IRS39SnoUo1NHpo8mx5vVa9Y+7zrlZr/3BbnQ9H0WeeuVeL20V/qrX63Wand3HY855aaNrKPGFylyf+Vm66bHVUe4bL0E7BTxSEf7NzomXN4p9YSG4/Xu7s+PqKepqjh+ZOzmccntBe4z2HVp4Hb7Oae5eb4IuOs4GHWo4WGfm6oraenUZoGgusD3U5MNbsfZz2O7lPmrxSvzkkdO7JSSYvNpbYR+aicOIsPPD/JwZ7yxbBY+0roDl6aIp99NPu9+1uN9MmgfigZq1tPUD6C2jHY0nDJA2O0wrNYLLrOBsH0HT9adqA8695TQJztGNJjekDuSj808+TrGSeCaa09t1x9xXQNJ6NS1dPGpqqJqg/jumrS94NJxc0ZdWS+ICWb2j2Eg8V0IxanO40+V4XbBhkbYoyYgFg/Td6AatNudeVEu+/6PdM8+YV23nFmkPqPjpuaxjQDL+oCFu8f5LhQDUWDvPsrIpMgDhthTtWZfeSHqrlc5ypje4kxE2T4kg7A0rpFym1H+1vUp+znLtI+yY0I9ddqB9mYqn/752pYfuZCk1KH7ku+yJtz7Fwu+bn/8Z4le+C0CzTdePnqtVqdyryi498zDNA8+xBU4WadtnS0p/FS33afZrFRQ/+IXcWSFcrWli0xdi67aHyZQ8qqbGvPjl6QJ3FL/nvB3V0Xm/ul+7aC35uFXM6DP7OWtvm8dIykMea7bu16t2ganSG8v88ETPHqvfk8eyHptf8A2Otnu3a0s5tHJPr8wMBNH3rzHcXF9T9O5LFF8IPLa2NILFEyU8cKup8ByacivYzCzQvdTq/52sFk9wbcb/9l26z15wZJ2jyF5bnu4NyvLG9aFkzQHPyoSlCknT+Gz7nJP/sL9BEaKoq0PvqOx+vWHN5cZBFIiYDt06N3Qg8XiTglhLj4hPO6KRjDiDbBORt1Vn9prmsgbt8NUq/dVsr8+EY2+CCAYsOeaAnX9AP/0FF3cm0cxdt34QmT7Tp38t9u2GXPUsbPLGk3Ysqm7vH75E8l3zBDNPPTJZmpzej+26zNntN9jFlrS1V0GGsoKnvX0WNWbPMWlZoWjytQaoAmsWgOex9trKAkft8YwA/Xhltmao0xV0SZfXIwqJ8Ma1YwFqxDLw1DrfoDgA9gXh8YOAbVpXqjx6ngWJOEDHx3M5q7IJlDCK4WTrZkDS59MQ141jib6y1nvfc8hiP/+2cPsbrH7OWN6Zpux9Z2xCDTVjnYfgP28+s0FT3NRZMtxwzdtA0PJ9QRj0rND053sPfyUmQbecOoJkNmssuXclzn9PAJ+Ps7Im+7/3iJ+nfXEqwAumWWZU5GiANYqCWCaAnf9rkkBPKtyojx6ky+oL4/b/R2yacI0mI+I1ji18PbNOf3LpEf/TmZjl30fazQFODL08bGppRqAzbz6zQTHpZBu6P/3zzQNNmVXDQR+FfdIOACU3jZfrCzKjDPZ9s91zMQfGSlOOJh7Dm5FwRycFM0Iy5VcHX7uLZ0UETIPj/MiwtUaWkAutETrLat367z/TvpSUwv2W2o9soYnmoEIJ8IVD6kwxD9JMmWc5tHLNW+IGlQDNPG0nQHLafeaDZtzbN0AG9zZY2PkobMxmttyMb/IusQIhCM2xpyEkjxyKgOYnQFKFFWtlNsFxfp7noRiKI9WwPTlmdoaUigyZA37pJX6YUgEJZlWIw+5MsiO/5oBbHRFzzoH3DGo1aKoPbbV505bY0ORnlG6eS9dzDtp8HmlnaSINm0X7mgaYcKzKZyCEtYqlV926WMTNK9zwKa5kY8j2RZv0aYpqTGdMMqjFF8i5ZXXRiusC27KdtsJtubvQ3wfEq+50+8fh52F7jRsNh1bl7hiv3kk/25Tq7Fo2bmlZNNOspYhWWbLgNFMqafmYCLMu5MxyzXhI0M7WRAs3C/cwLzWDQTdefui69bb7sTgM0VXgo2LE1hFWMmOYYxzT5B+lsz99YqTPQRQ8lDJTFNRUPmMYh2FFv5Wi8M/isqHLLUt1lbVUy+sicZDq7Tmu1b+wWsFoi1U9WBQmbpAkhCh+w+rr5htE7oczfZDm35ZhzOvaW5jJmgWaeNpKgmfUcOwXigzZohtqLeBqnBZqhxFDBvsI9H1/3XM539izJkuxn0TkL7fvRw9BU+83b7e4FHRyVf6f70clhrp/Uy5PEekBuOWawMmOwiEyyfkbYfi4jeH+k90Db1nLaLFu+BEatRVVZ8fBvspy7SPtc/B4tOOSBhBxPbLwWC9JvNmeumOtks7axE8qGi0TJVJ77ZG5sYPXla1meGW+z2228wyh9rFYiTEWe54H53Pjx/mBlOtbNrd68RTDKgKZ28RbEsrMgqVRJHFuA5kRBM0j2iK94tn8dbYf/Nx+nOrbNmp8u2sapGdMUC9hXWu68SgAFJaiSyj5FMu18cfv+rNv6LM9ACwpARDK5euAOsli5ez1P6UOjwMid1Iy92/qk03F/F1wjX+Buub4s5zaO+SV44aS0H/maoHAlOlEXzygQMagN2/ls0B50Dn6f++dIj0Ontaks0If6JWs/Pn8RDO4GD1ts1+ZKJ1mcItRA3YeA5mRAU95n8p/+848bgr1wURe1wqS5HYPrWX9QkOla25OBEKA5Ke55GcLAhYhelzjrdv6E+wFoApqAJpTibrdc9xO9zRIaf2ielsr/efsJaEIQlAGaH7z/Fnnrfx/0eu9ngZDnNVi9ztZ1gZZxhaXfTydvP//CfnWb0CuPAU0IgkqBplgOU63+q1ol/x5UY2Hkbrbs5z/z9hPQhCCoVGiG3Ngxhmakn0eAJgRBgCagCUEQoAloApoQdDqg+V9AU0Bzy4fmd4AmBEEDoJl/ydGkQnNUn/AFNCEI0CwHDkOWYwM0IQg6U9ActuwdoAlBENxzuOeAJgQBmoAmoAlBZwOaa+N8bQY0M/UT0IQgqHRo8qRLULC6/zXFytiBxe8n//S28Q2u1H4CmhAElQpNXhi5SsjP0eTMsMWZS7cwC/YT0IQg6Fjc80kVoAlBEKAJaEIQBGgCmhAEAZqAJqAJQYAmoAloQtBEQrPR6UyrZTmhPd2TeM1Je9q9Ol0DNCEISoUmC32P21T829wTYVG6dCVpT/u4LZ8CNCEIggBNCIIgQBOCIGgs9X/QWwjuzYCdTQAAAABJRU5ErkJggg==)
us, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.
Ans. Hence proved
Question 19.Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
![](data:image/png;base64,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)
Answer:As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARQAAABgCAYAAADctI0aAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAvbSURBVHja7Z2/c9tGFseXbi9ylczkh5ed7ZlUEZZsMy4y9MqW03FkCKPmPMcCo8PEkmZYKCfKnSeWXZ9TKa7uGpdOJvK1N7buD7BvrHS5RtK/EOqwCywEgiAIkQC4S32LbyMK5MNi97Nv3z7sI48ePSIQBEFFCI0AQRCAAkEQgAJdlo5FyJVRQvsY9yxrGc+zBqBAparXW/uMEfKb39n+GBY95l5vEe1kjlxON9KfJenT1vomgAKVDhSLWAdrvd5naI/51XqLbgIokJFASbjdNYOXfrV5sQVAgYwDiuj0nU77qs3ZRr1O3ktXm9n7JrTDttu+YVFy6Nt8JkWt/7RW3OVZQKUMWwAUyCigiM6+bjdtSsgJoewNd7w7ncePPzIDJt9Y0u5EzEEMZmb37uliy6L9t28BFGjugSJh0qJbMvDHvQ2T7v/x485HnJIDyp0dBcCeZ19fouTXmhzI7Le1bvdz020BUCBjgNKz2T0TYRIMWL7Y5N6DUYObEHpa1U7X7ib/qixbABTICKB0u2ufy21nyg863e7CPOWwrDLyk+8VHOmwAzatLQAKZARQREcV6/smdx7yRv15tOan1mHb7d40tV2kV3CNvCbM2Z8HWwAUSHugRB3dd8V9oHyn1v0PHb4UBBbp6TfutmViuwTBUfZWB++kCFsAFEh7oERZttf46+SOjhdkbJ4Ra/VFlfczJvU8V26H+Psqo0+Zs7s86+cjbHEY3ZvWFgAF0h8oHl/0PZFTwlZ/GgGbI7kzUeEs741OPR9SWhA53LHa1CHAXKQtAApkjoeS4oWcL4eqBcq02nXYctpOi+m2ACiQGTEUuZ05nB8RAUXs/hiQ4Ca8ARGrYMzZSfvMdFsAFEh7oMhZdI3dFbs6Iosz3tlVxqcOcYg8A1jk0lBr9VkyztJ12zcZ40+qgmJZtgAokBFAEbIZ2SeEnrTs9btRx6fkrQnv8cReGTiN3p1JKDkQTbQFQIGMAYqQx5sP/IFwrM5R0SUOMdZup7mSHbyt7kyYMm0BUCCjgALpLQAFAlAgAAUCUCAABQJQAJQ5lkj4A1AgAAUqRCJJLpnRjIaBABQIQIEAFAhAgQAUCECBIAAFAlAgAAUCUCAABQJQIAhAAVAgCECBABQIQIEAFAhAgQAUCAJQoHKBEisxUNPxplNKIGhpJ4ACzRVQJqk90u12FmzONsTxcjqe8xk7/k6d+nXatNfteXiY4fOq5fzfSmFaNlDy1sKp+HlcMb1k6kWdg0ygyA9lNfaBI+LOUv7WV/Bw5Lmf8vMzHYGy3qJb6oT0sN7LSZWFqsuUrEubUgwrdXBTcijum9m9e6YDZdtt3wjuJzwTlVqHrRV3eZZQETYxSt5EY8Swkqmi7Vy3bbVabIdS8t+8Z8yOB0qsPEFaBTdV4iAOD1WHVjegKPvjjSNOU6+T+jvTgRIVE89RljNshw+yo+cAkM5AUafeJyc42f8qguXQvfoTVaPBf4jGjWdfD+AyXAZE26WLqDJQr7+r18n7ixxanQmUns2/jg+0USUhRQNyu/e17kAJPa7KThev3PMKB5J/f1u5IVS//aOpQFGTGeXOTnzwLlHya9AW1Rf/kjYx/n2yTVX1QxPKfaSMmX4xHornfToAjhFAEY3oebufpgElEYep5VirDa034+u4UWu6rOvVelbUIVEDbszvXBnzGxPbUVonvkZeN9fWHsiynDm8jqpLYZYBFDFI0069jxdX18XzNKl+UGlAGbVkGNdhI6D4LqfnNO+HAdC+KCrU6XYXhmfKzgJv1J8Prje3b6jPZJA3XMeF33cStyHr+tjscKxm8PMYEPugOngsmKyCtSdxeye3o/x1s+yslP9T2BHEsMYve0Q5iqbjrVS53Kxyl0fGkwg70mFXSUwstrDHgPpBWgOl3uDPufNwKejAsrj00DJIuN0NSl+p/5PrTVX82l9vxoO8dXb7R9vrXQ++K4DBuOvzLHkkCBLlLMPaJX3/Xv8loFKkHWUMHuVpqHtMqwM8bZDuojt+swJK5KEwZ3/GIBGBTSYDxoy/TJtMAZSLeCgxeKg1ZNIwORhiSxChMB4QDfxokCQMzXv9OKAE/z/sHisIqphEUXYUHYyND1TVzkXHD8K2+COP0pZRvd6tLz8mH/9+q9f7sprlBXs7S+9EQO12EMuJgsSjPPS5AsqtTx6ST279Oz6plAqU+GA8X+sObUMHCmeYUSDIe30WUKLvSLmnyObws6LsKDwY63sjCmKdTvtqaMvMdjmmWfJM6wmJv60y+lSXWIWML/rebriUzh0w18Wz1N5Dif9Y2jZuBiWHBnLe67O+J+ueos/C2b4oOwp37dM9hcKXPVUAZRpPqOpA80UDyMH2drU7T+6F2tPdmAugpC0hVAfJA5Rx108KlPMBmw8oee0orJOKXatY3GeorQvsvCbEUMTz0bnWcRCHq34re65jKKlASQlcyhwJ5vRyAWXM9WOXPJQcpO2MnIMie+kV/t9RXjuKDMamufbn91Tcsmf6GEp5QBEAE3ETv513qoL5hMvTzbQJAEApCCjn1A7SpcUWazygqa7PSkjLc/3AjJ7yPeFg6YdB1dpgcI+csLXdu0XZUdSLk4Ft1uGoARrf7dFhUJUFFHFv4rmKgGfSU+q67ZuM8SdVD+C0HCS5kyiT8PRbjuUEylahQAlnASaBIgZNt/tFWkcVDbdqkRexBLKaeuiyg4fbsMOze5QjEuaJBNux4bp4K56Qlu4dpF+v/k8EK5VdYpC1O52ryv7YjN4XLwpGwU3xt9A7yWnHhyw74glXrLX+l0kfchR4ZfyluI9RzypY9pCz8J5q8waU2Iuep7E2H1DVca0oXUBOLC6TYPPHhMPoHmXOnkkwiSWD9ol1/0WeiTAXUGIDLnP3Iu3/BJGH3OWE2xekS9N/qM9F/or4PP13B0GRdX3COzrOsl/8lnjo55/T4yZ3vivUDn9QLUVbiZMlXQ28hxPamdzulu9hVLjTNCughLlCGcuv4bYpW7KvxJMbxRigjVfc8e4Y5Zls8q/SxgxtuZtTAwUqI0DHepfpbBCch3I5BKBU70bWRr2DAqBAAAp0IXfY4fzPKj0fQIEAFAgCUCAABQJQIAAFAlAgAAWCABQABUCBABQIQIEAFAhAgQAUCECBIAAFAlAgAAUCUCAABQJQIAhAAVDQMBCAAgEoEIACASgQgAIBKBBUOVDynLBvgoo6W7gAO64UWYsbQIGMAYo4dFqW9VRnxlLrsLXiLpsElbA8qdVqsR1VJ3tWtmy77RuMkjdF1uIGUKCKgCJLkf5v0lKkqgJB4ozTvm4VEscOOHHeb73+rl4n72dxiHb0PDy+2GjwH9SZx/JwbQmX6Wpx/5X96Smh3/6SqxQpBM0CKOqQcMqdnfgACA/87ptWPCuayWcEFNmejH8/qkDcNGVcARRIe6CMOoM3Xpak6tPtTQbKOC8QQIHmfskzSqJ64qQlSQCUgdjOFVu0JbP3p/keAAUyFiiRhzLjmkMmA0UFiWWwm7VexovwASjQpQJK4KKztybmtugAFAHk21HhuSDATa37z6aBCoACGQkU0WFXGX06zXq/CBtS8mJGqabrkkeARVRhDCsFnuWtYwygQHMBlLDO9OasC48PldjNUNJWHYOyYfnRk2l2zQAUyDigiMFoeuVFHYEi5DCyD6BAlwIoopOKuAljzk7aZwDK9BKeH6H8IJmjAqBAcwUUWRPaZveotfosGZ8QKfmM8SeTDoIZA2VrFr+f9u5Ot9tZCBII3YmXkgAKpD1QZMzEbtoiizN6jych3Wb6cYNZwFHabt1/UfVLgjLNnpCj4N0dl0ko+zBxGN2jzNmb5rsBFEh7oIgdiOygJz02JVM2TG8/Hhe0LVMyf6dR//vA79PGK+54d6b9bgAFMiaGAukvAAWqEii/AyhzD5Q9Hyg/AyhQyUDBiW2XQTI/KBHPQsNAAAoEoEAACgSgQAAKBKBA0MWB0u52v0h7eQ5tZJZGvRjptugWgAJVAhRGyAfT80igQG7Gi5G05QIoEASV5M2gESAIAlAgCNJO/wcMs4GZPtP5BwAAAABJRU5ErkJggg==)
Now, on adding A and A’ we will get,
![](data:image/png;base64,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)
.
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![](data:image/png;base64,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)
Now, on subtracting A’ from A we will get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAW8AAABgCAYAAADSHfN1AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAA7QSURBVHja7Z2/cxvHFccXVhmTlVxY4aKyrYwq4ZboVWjAo/VjxgWGOtyg0gyKGwQTEvRcQZOgOjqm7NZJY1pVRoU7J7GptIlE9f4RU51TmNSfYEu5twcCRxA4EofD3e7iW3xnJFEjPex+3wd7b9/esgcPHjAIgiBIL2EQIAiCAG8IgiAI8DZ10Bl7Y5QwPtrNZSFmPgvwEzQtP2HAMlanU39bMPYimIjfzoof2a1OCeOkjzybrwVz9+uQ+XzFK812Rn46hJ/MUNPmqzF+Wo8CHAOWA7wtZu3XO523MR4GJ2GFt7OCN/xkvlrBIoH8BHgbCO/Tj1uPtXhczqvMkEU8usMbflIrHsDbQHjTZDYa1XnHFmvFIvtBPmKJ2heqj4PvVa9anB0E8b6W4tbzyop3O6+ESzseXeE94KfvQz85e/r5qdSdv9e5+amUYjyAt2HwpolsOqLGGTtmXDy13dXlxs7Om6qPwYZ305IxR2p6wTLvFZm85GzeNSEeHeEd+qnscFY4Cr68Aj+13tfIT0ej5i/rBUEYTyHVeABvg+BNk+hV+LrcyLC9NV0+/85O402bs31uu1snYOi0nHeXOfs2NLh4kWX9th+PcyqepWg8vn/FdHj3/VT4TTc/0Vydmr92/Z1J5y/teAKPfTNJPIC3QfDuOOJOMI8BuFvaJBppu21fL9ut+0MhusCeMMZfZtkhERtPAPWk8egG776f9AF3f/6asfN309uwso5ncHU9aTyAtyHw9v36FdluyO39hu/PmdLXWxPsy2BlcqhK58Qk8egEb/KT1fVTtdGYD3x0ySg/ZbjyjpMr2F7SeABvQ+DdDMslr8u2u2ovFv/S6wvl1kHV86/qOC69lbdw99SKJ9lmnU7wplip/HbWTyXpp7y7NhLPH610g/lTIf5uOSVxPIC3AfCOlheCZPvTSV1t1bWXw023bB8T01K4ySOeqbLqnjQeXeAd8dPxED8ddf0kNPXTU1VW3ZPGA3gbAO/eCc0F+8lgJ0ArPO33mlm1R1l9nnOO816oz5X+vCb4Q1HfvqXCHKURjy7wjvMTnR6Vm2yBnzI86p+KnxzBd4W7fVsVP00aD+BtArxbdilYEb1kovblyETMsGMj5nj4GQ3bDJPtaQQ6RTbK0opHG3iHfjqO8dNhlh0bMcfDL+wnCbrgZyqUS2QXTwrxAN4mrbyHrK77j8DZtttNom1X3B7W7aF7PJqtvA+Hra77HRK6+ampmJ/Odp8A3jMI71MJNbAa6sGbulAUP1xBJqQ6oBDu1rCf6R6PTjVv6rFnzDLFT5tn3r6Xr59SiQfwNgDe8tu8Lm5RLXLwtFb3ZNexKrXjOGNTXzG37n0abUsj0bFiIexPsoTFNOLRqdvkXD8pUjtOOn+Li/afVfKTjMf35wDvGYQ3qSbYF9Qh0H3/xiUJGc6eqf5ek8iR/pe99z4MKMuDRxeKJwGEdTukQ34qMH501k9qv9dE1pTlkf74+ctyw7Ubz/HoeLyx4wG8DYJ3OKHl+/13OvAjlWrHI2N2yyvxG1LZvoN6WvHo+G6TZuCnImO/hOOhh5+aivnpIvHghCXgDakMFbzPG0pzkQF4A94Q4A0B3hCSDQK8IcAb8IbMSzb4CQK8AW9II9HhjGGnFuEnaBI/Ad6ANwR4Q4A3hGSDAG8I8Aa8IcAbfoIAb8AbArwhwBtCskGANwR4A94YD8AbfoIAb8AbArwhwBtCskGANwR4Q0g2wBt+ggBvwBsCvOEnwBsCvCHAGwK8ISQbBHhDucP7sbx77fEbqn+4ceKM3ClXMBHe0Tvz8vqccfHomDw0huOMpUnwhp/U8dNY8K5Z7BFbsJ+ofnv0ReOUZufsgDH+UjidO6bBm+4eDD9f9848bj3v3kuYS8JteNX3BGdPe1dDceug6vlXdUoyz6uKSkVscc5+vOhrXk2B91k/lRTzU0lHP1mBnza7flq/yFiODW/fr18RjL0g0CW5dy0rjRMnmT34uz/Jic/hS2mayRbe9i0vPu3dmUe3glPS0c3g2SfaTSt6c3en5bwbJp54Uff9K1o8rtbFLVYsfl8ssh/GuYzYBHh3/XSklp+WP+75qV1/J/STpY2fOqGfvov4aTrwblb4enfS5H+i6oAkiVMCv7j0V1PgvbPTeNPmbJ/b7lYUlsucfRsmXADMDOuiMh5hfzQ4vp2WXaKbvoW7fVunx1yZPDME776fnFN+Wor6KUNgynjK9sYIPx3n8RSdgp9eTQXecrAW2JNyvX4/WKkeqlo6SRInDYC8pspurZlSNtlu29eH3fZ9Mj70VJLlLdrxq7nCsahv3wK8VfdT8/4oqCvnJ3dTx8XAdOAtB4Uv/41A6Aq2p2rpJEmcLbt8v+y2VnJ5bMqhO6AmWDDp4jDvjgTaoHEoFuHs6ZRoswhvjfy0R37KexNVKXjTBJ2sTE+My6zaI9UGYdpxRnaFL6KCasnWW3kLNzdgUicQbfrJjS9R+arh+3OAd8rwzqiE0feTo4yfqg1/XlM/pQ9vqgdHAXNSp8y6bnqesoiTLv6MbtjEidve2vnJduPaZXb55xudzrXsHivFs7zmjZKd6qSRcXrNrXuf6gbwseF9461V9taNf08f3rn46WmefrI5+6bXbSL9tPKZbgAfF96drp/OhbfcALTuPTpZUTYa1fnKAvsXDdRFNwbSXrFOK05VyyZpjB/9eU3whyrUlynpWm55pdu9kMsG+CRjOs2V9yRxZe0nR/BdNf3kfZh16SQFP6W78vb9xtzSQuHJiNXlhUsS461Yx984TCtOVeHtheP3a9IVf56bsrEmbNvXw3bG7J/iJhnTacJ7kriy95Onrp8ybhds2nz1wmNa8dpR8E4F3h1H3GHc3h/V4qVK6SSrOHWtedNkD+s+UUHhxrJaJbjUyyaGbVhq4KdDXXq9pwZv2gAc1oPbbxFSoySRVZzp17ynm2w0uVSXFMLdGvYzFYwry11DvngBb/Xg3feTo7Kf2tJPGu2jpA7vcDPCOhhlhH43h6wz5zZxusSZdbLRZ910SndpQzD49aXoUwEdcxbC/iRLYA579wSVu8LDH2o9fo8B7/UL/33N4U1+oifcUX6Knp7N009LvKCrn9KBd2+zT9hfVRuN+RHfwCIsSdCqtlnL44VVusSZdbLJmqQjaiefe5iyrH/Lo/B0aIpbz6ueJyj5ae5cwXe5cHZ1SjQCBn0pRhYEl85LON3hHfqp7MT66YJPIen6qfT8g0ZjcdBPOvV5k582qOxLJ1XH9NMZeJ9634cUPxo8PSXf8TBYKhC1L7IGnw5x5pFstPMev3lydqymKVm+Wix+Ho2pWFz82nZXl7VaIbXt60XGfhkc2/NWerrDW0U/Vc74SQR+ar1vhJ8GNjTHLptAepdNIMUei/E+byhlPwHegDcEeEOAN4RkgwBvCPAGvCHAG36CAG/AGwK8IcAbQrJBgDcEeAPeGA/AG36CAG/AGwK8IcAbQrJBgDcEeENINsAbfoIAb8AbArzhJ8AbArwhwBsCvCEkGwR4Q4A34A0B3vAT/AR4A94Q4A0B3hCSDQK8IcAb8IYAb/gJAryNh3eS2+tVFMWtQvyD4zlr8Iaf1PcT4G0AvOkCWIuzg96dgtx6Xlnxbmt2l1/B86qiUhFbnLMfs7wTcVAbXvU9wdnT3vVU3Dqoev5VbeDt+1fS9VNJVz9ZUT/lFX/XT//p+6l08EGj/QfAW2t437h2mV3+341O51pyY9wM8owdR+/AK9BlpkHSlZzNu9o8CtJ9o8Xi98Ui+yHrC20HxzN6E7q87FaCXLxICsU/it89ZPzuPzXy05HufuqEfvou4qf1PODdadmlxcXljxu+Pyd/366/E/rJelH1/d8n9NMu+Qnw1hje8mJfzva57W5FYbPM2bdhwgXA0az+KVcVOcFbjqewPzoZy2gC0s3pwt2+bTK8+35yTvlpKeqnCVf1OfnpVR7wluNZrmyM8NOxcDp3AO8ZhTfdPl22W/eHmmaBPWGMv8zyVm/d4R2/Gi0cmw7vWD8FUNfYT6/yWnnH+OlIuJtJ/QR4m1A2GaWaYF8GK6VDrLwnE20uOTSWwtlL+m/oVDYx1E/KwLvrpz3yU9J4AG+D4d1beQt3T7cxUQXej4Mkow1UuXEnKl+d1CxnEd59Pzm6+il3eA/6qdrw5yfwE+BtKrzDxzLxTMd+XxXgTbCiOm9k0+41t+592vB35hImm9bw7vrpqcZ+yhXe3bLTN71uE+mnlc+SAhzwNhTeNKE1wR+K+vatnB4LC0N6hEepoHLZhJKu5ZZXut0XsmNh1uBNc+QIvqu5n5Qom5CfmoGfioz9EvrJ+zBJTIC3gfCmyWxWeJvbrbW8PpNn87XICiNW3PbWVIZ3L6a2fT1sx0zWvaMrvPt+8nLzU9Pmqyn4SakNy66fjpJ27wDeBsKbjDqsWwBlk8nl0ibTjMHbID8pBe+Inw4B7xmHN00i1SWFcLeG/QzwTmEFWOHrjNv7gz27JsK77yfHFD8pB296opF+SrARDngbAm+awE2ndJc21IJfX4rWAOmYsxD2J0mAowC81/P4/4e9e8L3G3Ph4ZVk5QOd4E1+6jjizig/RU+fAt7J/bTEC9JPqHnPKLxlTdIRNTr913sPxYDyrH8nMTp9EcnYrXuPuvDILOHkUXjGDundMFXPE/T/NxrVeVfwXS6c3aT/ri7wDv1UdmL9pNgT0Xl+2gi+iAjeOfrpJ3o3zAeNxuKJn5yun9DnPcPwpk6I+M0cfqTLibj+Js7pz5PlZpls6Vosfh6NoVhc/Np2V5cn+Xd1gbdpfup2dZzxU1YAJz9VBvzEufV32229P6GfAG8TyiaQ+jLhhCWklJ8Ab0Xg/TOSbSaS7R/wE5SmnwDvXOGNm09mQbJXOoNaMfw0G2rZfG3w/eQYGMAbArwhwBtCskGANwR4A94Q4A0/QYC3SvCm65AGD0RMekkplL1GvUTJq/D1LOFdbbUW4Cfz/QR45wxv2cQ/tLdWn55aKFTcS7my6FcP4c3+Cz8Z8sQW81IuXvEAbwiCIO1X6RgECIIgwBuCIAjKQP8HrJ4BCTqS0N0AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
⇒ N’ = -N
![](data:image/png;base64,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)
Now, Add M and N, we get,
![](data:image/png;base64,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)
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)
.![](data:image/png;base64,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)
Thus, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.
Question 20.Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
![](data:image/png;base64,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)
Answer:As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQ8AAABgCAYAAAAD+vcFAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAwxSURBVHja7Z1Bb9tGFsdHznHbnlqgTTw8bRqgp5ojHVvk0Mo0mvgm2JTgywbrA+EV2tiADqkt5xY0bs9Jemjiw+7Ch9y6WyTpdeEmH6C7SHzrHmr7I2Sd1RtyqBFNUbRISUP5f/gDtmVSj8OZH+e9eZzHbt++zSAIgs4qNAIEQYAHBEGAB1SUDsTYTD+hfQp5P0sJ97QEeEC5qN1e+UAw9qrTqf53WvzQabbn0E7Fkufwm5379zrmnp7w6to64AHlBg+b2U9X2u0P0B7TrbUqXwc8ICPhUWSXZ9D03mT3BPCACg2PW17tQ8HZfjhd5vbzmte6UoQ2aHm1KzZnzzt2v5Hi9ovqknfNdIB0YLDBmDhIe+8AD8g4eNzyPrPLZeeb1Tt33pLnbLqXfZB0OnarddHk6yfbOWNHemyg8zg/IYjMuZuLhtt9DHhAhYXHnTurbznC+VqBIzxv05mjzi0a29dMvXZpO2dPueNu6eCb5+yJDxAz4ddqrVzswPmX0EbAAyp6zOP0k7F0JFa2vzD12rfXnY8rTvNGP6gwxo9NW20iV6ou+LfcaWw5s+wZ4AFNFTwoiOcK9ogJ92FR26VO9jPxyrRVqLYrrnPR2JGAAzygaYHHXgcanlcTMvgoqo9XW623i9gm4cA0DH7krpR5+a90rwAPaGrgQZ2ZYgVa4PENt5e/KyJAfJdL7Js06yB3pSH4jnKjAA9o6twW6tTNRmWJM3YoAVJd2xjnAEtKtU+Tx6EG6ahjNWe0dUYOfMe7eWp2BHhA0xTzIFEw0l8CTd+5syohLfuU9IGoD+joIDXBVuuTTx7QLK7z8wUFk9XV2jvzEh72Qa3ZnE2TLAZ4QIWAB6kh2MNxwiMz8BriWtzqy6TVDzQqH8X/3X45qJ0BD6gw8JAZkNx5Gs0BMU0046A4hxCNrbjPTLQZbgs0FfCIe5el1Vp920/AGr0LkBUccvkz4haQKG1dCOeuifADPKDCw0OmotMr/dx+UfM8QQOQ/HEKOnLh7pgOjjW34vpp3sF7LRFFBxvgAUE5wUN24rJ1T/fJLav8o9P4asH0a6dVoeSgpbn7mXSXxgfHOgAPqHAxD8gsAR4Q4AEBHhDgAQEeEOABGa6mw28CHhDgAZ1ZlATHRP0R4AEBHhDgAQEeEOABAR4Q4AEBHhDgAUGAB+ABeECABwR4QIAHBHhAgAcEeECABwR4QBDgAQEeEOABmQYPbRv4kokXHbNNvZF2Ah5QYeExTN0K2mvSdcRNU4sSa1vFHQa7Ox1X3DW3aDduT7b53kzKax4bJPOER7SPFen+pK3rYqitM5nhIT9kpZPIdmpvon+jLdsVKOqC/cCCLdxNhIe+A3dQef3QxALEg1S32S6bdZ4N2kxXDmYq19i5RuG2rxcFHre82oeCs/1wWz9uP695rStFuDe00bHf5sHepdx+UV3yrpkIECo5aTN20LvXKj9KMx4Gw0Pb6p46hqAv0jqtqgCug4I2CTERHsp+fQ+C7RXxhcWsX4sED7rh8j50gPCZd8tOcc0v5QBMARsT4EFlC8pl55uw39GmyBIk4mCl1bpo8r3xS0tScarT9VDm3M1FAx+m66f2XU1ZUzcRHm3X+VQfVHHw8G+uM+e47U9Nh4c/kzJ3B+szzZ5U/daU5RclcKz5B6bDQz6MhPN11M5glmikK9xjuywP4W7p4KONhX2AmAU/v098/n1tdfWdYWZFyTOPZvP9Hkj0gYesJ9rcfj8OHpG4SWkY/1YPwHZ/3ps5i39Mf9t05xbVgBvwPbH25mFHLh10lj2rrKzckCUKUswmumUPmyOveTKqgKn/RC8djbrma6aH07rzcVyFOAUV09zjumCP5Hjg4qfArZrJDR79pv2DOmwIj46P3WxUloPg5Am367FVzqkuh1O27ofTJs2/DQOwnP2HZgzB+Y50G5KO155ah92YDQtiNiLcZp7OEQR6VSD1iOwlKg9hx8j8dDmI+MLf6Tv98ouDXZemU7lRaTSXxuUa5g0P6tQudfSU02kjY1RyoIpXpqxCEegsxn4PxsSJis3UWq1LRsDDKjv3Vb2Npl8f85QrQ1OnMuf/UP8XFv0JpnhBAFYOekvMP3Cb7cv+ufyBP+j4NG4LgUEW+dXiO0ENjhM2W/2ZgJenHVk7oZpBqOthdn03x4GaqTJ8nvCgFSXPqwkZfBTVx3EPniIoLKhkIPw8z/tjw6l8SQ9N37WyD2qt9qWJw0MHhfJZowNXDgbNjSAFPn04yMNBEjE07fGD4OH//+kppQKeiisMtsPb0EsMel07NvK40X5kvDswgzY9yhNQWSvDt9tXP3qXvfvb1Xb7o6wDzi9C1F3loxKORQSI73KJfZNzX1R7l4KHUZoYyPbV975i7139l/6/I4WHPvC6Je7YSWwHFfUfkgZ92uOT4BGeI+aaQpuDz/KyI1Og1F7eVYAiN6k6y35WLqIJnTBvt0XG0zqzwMCdzA3E45iFqXNQqcxRx2rysLW7MpfuYTT2mYf+ZXFLp32NjBm0aY9POk/SNYWfBfU7U9gxso7tu1alZ31mAbm6LiYGTMlHD2dZY3qCZ52FdQPVoy/OndXW3tl2Dnke44JHv46vCDlo0A46flh4aMV/X6WBR1o7hhqUrriux2Qi7Zqb62JSzCMqP0A8PnhkBl6nv8Stvpgsvz+JX9K0sQnwkEHFaJRXrkGLRjsFPAYeP9BtCZbRoqsW4fWKxsOMdmzlESiNy3Ho2p+P65I95jE6eOjZwSYPQIIqxTni7rvp71H5iaELT9K08UTh0X2aBMtEXuuKHvRUxycld6U5vufpHXMemqpRpDkIvJZ6A13d3IIz2nFBBU11O/aGeNfBt8N+3m9Adlddlncn3TnzgEdcjgy5bX4C1uhdgKzgoH5GwV09eE6itHUhnLsmwE/NMOPiM2lzUVLDgzp9pxMLCY9gPTjuxSwK4i3TexfdZCyZUKUStGjpU+VO9AZpgrXmMA/DXwLd61092Yhe8KDj4+yiQeZn1fn209N7wY/snwh3ra6CkfKJHsw68rCj6wbxY1Fd+3OaGxQGRUX1sd5ukaeckGCWsw+yf2+mqPAIl7glhD1BA5DagDo1F+6O6eAIXro87n1XpCtTspupPS0x/7160BGc6W9nyQdKBY+YZbPYVYS4/6OchGDJs3scn3/Sm97uXl7g/G/qc6vs3KPP47+3FwpJx0dmPYdJ9tN3UeN1P+eHFafx5ZB2vA7suB/N0A3O8zpNwpAGpNfKpuhTgd7NOf1uQj6rO5OAhwRs2bqnX5NllX9U+TMmK8gNSnD5Tt+/SdoaJIkFfbV6n3KXzuziYDOgSZBftKdxzwvs53F+BHhMwifuzISKFoUHPCDAY4KSrpHj/KkIU3DAAwI8IMADAjwgwAMCPCDAAwI8IAjwADwADwjwgAAPCPCAAA8I8IAADwjwgAAPCAI8AA/AAwI8IMADAjwgwAMCPCDAAwI8IMADggAPCPCACgkPvW7wpK/ZJFsSbExVuwXwgKYWHtTpqQRltSq2VN3gSULDFFuSRBsvy5Kdai9Vbr8ICluXAA/IIHjIcpP/zVpusm+HpT1bLetXy2L/nvSGwibZ0k/+zvu9+/eWgk2659zNxSzn/ov4w7eML/6UqtwkBE0aHj1PPUMGrEm26FI1fbjjbqmNuGmT7qAm7UnWQmGABwR4TCk8qERn3F65eqGzLLu5Ax4Q4DGl8EgSVSBMU/4D8IAAD8Cjd+ZBxceE+xAxDwjwADxSyw+iiv2sK2KABzQxeGj5EWlUAjyyS9WjVfWXAQ+okPDw/DKkCeUZ9bKlvQWuAY/hwEFF6PMqFg54QHBbzgk8yM48KxUCHhDgMeXwoMFNcQ4h3K24zwAP6LzCY8MgeEzcljg4tF1xndvL33V+vqDHkShtXQjnrkogAzygqYcHdfxNd25RvqdhL+8Gg6I0ocFpjC2xMQ634nLGjsP3WiLKMlsCPKBCwYOyJi3Gfo8GWvMKAhbVljg1G5Wl5IA0P0SGKXTu3BZo8gI8oFHA4zfA41zAY6cDj38CHlBO8MBOYudFMmckEjNBw0CABwR4QIAHBHhAgAcEeEDTDo9aq3UpmpREQhsVT/1eYPSqfAPwgHKFh2DsZXx+Qba8AmgySnqBMZrTggaDIGi4WQoaAYIgwAOCoLHp/9Ijq3aFvm6hAAAAAElFTkSuQmCC)
Now, on adding A and A’ we will get,
![](data:image/png;base64,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)
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)
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![](data:image/png;base64,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)
Now, on subtracting A’ from A we will get,
![](data:image/png;base64,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)
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)
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IK0PycJvMcZ3fCNsB5EOFzd0Zha9R+EG8ZPGEvDSACjTVEeudB+pdJO0bt4HHqj5PhfozEMWiJE5T0LdeXP3S3m8st+7NilHtanNBug8/28kN2RbpZWi0SD5O8H0AEGmuIdM3GETtMsJdJjQrtvTAsgDQFv8rF/JWijcN8hX2tML4hW46yabv+ClfWR72nxfLKFulz5XkZNt1uVBuUfE4DLkAEGmuI2Nvee3su7ERmqSS+L3JD9QW9+pGzevScSiekKS85lM+Vq1ZP5AE89W7D7B5J8zpABJrYnAg0HAEiUJEg8jsgsi8hctWCyA+ACJQzRFDZbL/KbgmiLSwFf47BgQARCBCBABEIEIEAEQgQgaC0ECmv1UxzJtirxpGCccpXUYcKz83yi4AIVBCIsAfhh9bC20xAmXsckYcK+ezgzl8MGgRBexIGAYIgQASCoPz0Jx9Qd/nY6svZAAAAAElFTkSuQmCC)
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)
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAh0AAAA1CAYAAADlAFmVAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAABNOSURBVHja7V3NcxxHFZ+VOULCMSHqPRHgyG7v3iBQlFiNjBOgqEUebelAHLZSU6opbBn2IOyVbymchGs+qogTqpJYRXGCVHCSCxeI8weAvzgT2/8Byop53dMzPbM9Mz37Ya2k3+FXTlbz0e/N636/ee/1G+fKlSsOAAAAAADAvAElAAAAAAAA0gEAAAAAAEgHAAAAAAAASAcAAAAAACAdgOkhOE4txBL9C30AAAAAIB3AXMiG73d5p8MvM+bcYp2tbegFAAAAAOkAZo7dTf4jp17/V73u/DskIAcgHQAAAABIBzBf8tHjZ0A6AAAAAJAOAKQDAAAAAEA6QDoAAAAAAKQDAOkAAAAAAJAOkA4AAAAAAOkA6YBOAAAAAJAOAKQDAAAAAEA6TjDpQCdT6AIAAACkA5gr6SAHOwzchts7vwY9SpzvuWsr/k4T5AMAAOAYkg58N2SmpONilfOGHn+27QbnoMM0PM5e5t7w2UltUtn0wk9ezL3jsxAfEZuDrMDMSIe2gNlALHKDQf8rnssvMMd5yHu7Zw5TkL2cMVpMAO2cvaVDmIBLl7zGc0Q6nObZd8J/T9mMfcdfaTK29n7/pZe+bK+fvYWf6FXGmfesX3qp/+U1xt5f8Xf4JGPY4OwVx2EP3WDYWFQ99fvdxxZl7gHToxcS5dDmHlCUDrIuHKkv9Q/jPjI5tqpPOs66SJEO+bZdG4UHfaHhIPtbeOeRWuQ2uPNW+Bv9/eAwFz5yMqvMuZGMORxP+KZr5bjDRTs65wuHrd6wdeIziXBsu98O738/o/MvmOtfKDqPyN7qcu1jvrn7I2tH2nTecZbdjx+lfJM5fOdtMc7B4CuFUZ7h5pPhw7tJ5MD0rMV3bZZXS6+TQzquhte9v6ikQ9l7LbJ3kI7FXKSr21z986NIOo6rrENaQxznf7Ff+fbZPzrd7inTOl53nP9G83EU+ZKPaO2hdYoz51PHWX7YWN/58VElHru99mkbXezk6KLb7z9mJh3M/Ug5JaEsx7mnOypa7FzmfKQvclsdtr0oC98w8J4WYyalLHc+KXM44dgvSsLB720Oh08eFQOgtEoVAjEYbH5N6oU9XOSJXmWckX3eEUZt0EX097s25POoYpHmHpB+meGd4IWTkPY6CbIGXvvnrCZeDg/YD178jbNnfsunFx3GWh9sDgZfy6xDt0InvE8vQbrzPaq6qNecz8t0MYx00R0MnhojqYkjc5/R3+xMpEM6drfhesNnFnHhE6Ro2fk4jr4UOJzIwd2VBGXxIwBjMvLetQrO6aJiqFXrRvLebOaxyEwyTnqOrbr7uun59bhzzSZqAtIBzPKt3+PO21QUftxJh5C1cTJkDVx2oVaTa5OMWBzUxl963YZIeRvWm2ideu04rEW+1MW+0MXPBj9xDsZ1IaL4oS5MJCs5KAieSJGLHNJBTi8Idp8wLXyZvE9qIHs59RZ7BXUY2RqNMsOWkRh23XX5bykN5DQ33ilasNna2lWTjIv+ZlElj69ISntz85wgWTOQVehuxj1FjOMsmaBkD2IsOekomTKsHt1RNpf3u6095pxXKbdbdE8T6Si7T3YsOfNwSSOXxtzsnuEaJSR1TIbya6fy4qmiWZMMRevPtOfnHLOU1a0vifOIdV78tazP2svqspb897h8NvZTQd8zlVU/Tsi6Esm6Mp2sefOjisxl4y2zRRvS0Wq1XpPEI1xT+oNW9tw80kHHkaOmdeo4kDM/0cU+pY2+399uZ+WyIh1jrC2HdOS+bXnDZ4Ne+2xUozBizY1XdeXr9R/KYcmCP5GbHqsLEXULvP6GzAnxf3q+9wznvaEN6VgJgu9SGijP4Yhr1/kb3tBrHDXSIfVtX+SoF5yKN/8ZpFjmQTomGWfgts+1e8F6rg3TIhAStCrRnYHf/WZod28yh93SdUw247bqrzsUJrW0RwIVfMZ2LMAeLLsbuzaLT9k9s6QjlXIy1AbRWOLriXnVvNn1B9/UakRGUbrxDqUbd2VOez/+PYquaTVUI3VsVRnUWGtRDZbIgYfPPjWWKC9O11inolnm3KJ1hp4rZ84/RN6YNT6jEC4dI2sFaLzsIe9s9fWC98Lzg2C56Px8/TVu/rS//S1t7XmvltSVyXw2967Reer+ZIvxOrnsfkKLMtlcJ7QRsrmszdPfWvX6X9U96/yHb3b7g8eL9K3u1fAuPWcv68GYrB1pt/ux3f5w/QqF0yNZ3x2TtSFkfXxM1lq+rKI2y3FuJ7ZXC0mMvx3Zx215bXa/bC0oGq+um06r/pq0xcanoS1+j2zRZi4S6aB55scRj8Z/usFg2YZ0bHXaz7d7W+vHJRrkR7rYctl5pYszwYA9ctJRb7mvq34R9IBMYV91rO6wyIBF8WDmeCp81MmJcER8422rSEfoLNQYTNEOqolgvPeyrYwFTLp0h8+sIXVlX4NCxzM3uJC8+Zt1ctikYx7jVKSjzG70iRI5wlGW2G1EIXPNZt+yskeR7uP3vGD4NP3Wc9ll20LnsntmSQctqj23/StT3xYR3mXsA7cXnJa6EfVPdx2neY9y0PEcy5A9miMmAi8WcN65pOSqKoPsMSPGcEeNQR8rd/gNdW06Jy6a5e5VJUPo0NYlOWnedLn7W0VasuOtev6aIFRZeTP62978uhp7dzB8Kh1do0jH1kW1Buj3r/PVN0kuWrDDc2/3Avc0j51u+p4yqlm73+75Z+laceF56MDzooDhuvkHlQbIk7VDsobn58ma6JDf7fo739DsdkRERskVkdK0rHT/PFl993TTIKuwi5Y4b0S1eCGpeizRJb+txlC87qfHu+myS9nxeg3nGuv4IhWk7rnEvberkA46NpJRjfVxm/RK1bTVYfsZG9IR64LsTeriq4820qERBu0Nc3s83D3eBCvnje2ebjDiN967bEs6VCGheUKljrEmHRGR+cIGZTtQpklB2JIOWiibTvMjdWz0XB6I87VF/rBJx7zGWfX5phewhHQoWxq3R6/UHkVEINoVJRwtFQFbNIKzuaceZZRv4a3f5UXApPP3L0Zbspe0VEBcP6P0no0MaS8LF/W5rEhidRl6l/QFTOyI00iYTJmlrx2NIXVcPMczqbhonqaONf1W5XwiZEnKJKW/lNM1kQ79d0W48mxOd8Sr9Bv3runHywhF83bRvCiQ9Y6NrNF4/ibsNjxWs9u0rAbSof+eJ2sB0fkb2QLVTAh7Zuw9m4iuzXiV/Ho9hrLFqqRDH6u+i2NWpCO09fPaTpFiP7PiP/J6Gn9cFx9mdXFopCP7JmhLOvRJ2Fn3z9gyOp1Q6BEXfbHUw/iTOKXDRNXxisLMpPfHkghBLjuf2L5p57FuWmyzDmwa1m0a56osCD6YZvdJQjyrkRcZTUpIx6T2qOuQZNLSj6Wkw+ae8bxxvQuc8et5IWitwNq8eIWOLXXPjGOiwkhVI5W8vbNXyhyC7mDyZIifUXRt1WMle20T6cgbr3LwZaTD9vzkWeTrz5Z0ZH/XSY3uiBUBnMT+p5HVwm6tSUeJrGN9OpKUSvNek9evK8c2yTyLd1hEY0hSgsIWQ0LSPVXl2jrp0Mda03ZxzIp0zAuziqD4GV3s7vafiHbojJQujhzpUOFAR+31ZewW97Y2yppGZUmHyenQYqn6WxwG6ajw0MeKqGK9WoxX9fLIWSitUxe+XMD2M+eP4jyuhip9Q2Y9zmJnW63RV5Z0TGqPaqJveW2PFm3GvZeDDn/BtuV9dM/9vHuqeSOKugQpIOJmqKqP7NymtkU6LOH8ePL21vpA9hpIUjFrbO2qVXSwRAaVflAON6/p3WGSDrWO5DlRwxo3NekoIgOPgnSI4sf1tld3nM81ux3Nk3Soc0V6Rkuz2DrUeLwN83hpu2cUQSBbvM3X/V7els8y0qHe5ul+akdLbLsLSjqqR1AOajakQ7zMky4iokc7WpQujhTpuBLlnV1Z+CMXe/k2XLMlHYmBS8YrcsX11TeKepHMmykaHHgpJiEdIryo9V3JXGOq1MUs0yvzHOcsScck9iiu1WSv0gKrrlX1Ozvme8rFIHbE4byJ6gZEDjt3LmuRiqxd6wup0Ht0LN2DyGSgpUHEglKSWsnK0MmRQb8nzVGKepmuvQikw0p/syIdseP2f101gjgL0uHJrrwPxsczZ9JBaTvG/l6Ld8UcWMkuxlsrHq94ltubXydbrCkiTCkBC+JhIh3Kkcc7WrqdPmPu9ZNU02HQhdjRInVxxEhHylCo4Ey8aRX3JDCRjth5hczZddkr+oJ21Go6qpAOcpwmXWmh4olTF7MkHfMc5yxJh8Ee/1lmj6Z5UIV0lN1T38mU7FxJFzamI370t3SzHkHEtVqpRO/hsTtdrnqg6I53q8N/aavPXBkyDlGkYVjjZqs1nlpZENIhdRsEyyb9zTq9knQrHn+eZVs9pyUd8Zqp1WQ8CtIh347d9/rD/pOyZsK8NTVnnoXj9QrHmyUf0a6ekc13mvJIh3yxiApLZc+nT6ZtAHaUajqMuogKmfOiVY+UdMS/ayHz6G3qYpJ/k1uyOrxzWZ8cNv0pTKRDj3ZQdbNegHlkazpKCkmlrpo3845JdoeUv6nPk3TMe5xVC2+zpEMv6qOdGgZ7fFBkj7qjSUV2MrnxvLHTPfUFLHtPRTrGHJUhzaIiIQ5rfkbbPFX9DNXnZGWInRZn13WSrhwja7Xet9FnIkOy8MS1Cpl7RmH1UV5/llzSQc83x5HqUZ9pz0/018jqb1uXJevsyshIkc3Jehhp/6EdPK7uuRPaUFEqU8mqj7+MdOjHqmcROrQ4ypIuzDyopY6rKCvtItFTeIq8tVjrXRXVTFIXRPQGy0VzRaVk5HgPMmOQv0lbXDHYYu0+93bsSEcOOUkVls6AdCw6LHTxYXYnUinpoOY84QLHhYMLFyl6OzLlr6lo5yx900NbRAnxB8zETZMHoNdYUH8AOp+2+NFWtHi/N1u7EewGT9AEoW2tdD5NNElM8nsC0JgHg+5TnLEPo2r5pXEmnHTxTMlI48mRcZFg40TjYlHe+bMxtBU+H5JbfXPGtjZh1qQjLhad4zgnIZXChqJW+tKODlSFtm6Pp/wSexx/Wx08teN3v0E9F2pRrYrvd5t5W07L7qnPPd55sa90owh8Vl9aX4yR+j5RXp+NOMWSIekxMbDshpsjw7bpnvGxBruiuepF2+pZc/33JJeyj6Z4Vs175MDoN3kspQYUWd1bsjo/eqPWzh+p8zX93R7XX/OOnv7TiV8/JAo9130+/PexdNOw9DcryOaaUXfkxvrgx/E9oz4pTuqe7IFIR+aE8MX4G0LWkUHWu7myZnYeJA4/WI7s9i9C9sbZ2G5jWRs5sq5YyLq3tyTqusSzD4/fSxqM+SuSPJEfiYheLW+eqfGeCQKWGm8ol+9vhuMNGqusdoNx7xVF4MgWayU7gVSqQ3ZejXcv1XLWmtu1Y9Dq3EIX0U4u90smXcSFpUIXFqQj8/E0rUJ7462y4+itaCwFkfmImuhPrz5yRs2CQuMV59Tr/2q7vV/Fe+3d3vNB4H4nCsfGx+YtbuNjTha2qCL+PX03glHGR/zBt0mg3jZNIWitOZRqkDP20bLdZCHLfbY24bVp0keTj9O7NhHpqNCnIyoM29fTZAZ73C+yxxRBc9vn5DXZ/bYbnEsq9MP/L2hsVnRPzX73dd2oRkvJ782Ucxf1IeE8iBtNtTrG9vHx9TP6Vtev1A1XyLDy3USGZq7eTDti4u2QGVkvyTdZzT6kDUVNCJPfW6t/MunK+nxtZwqF5Mf0Z3D+qvGWlDVojI1fIypGm+skIXPRO4Y5/9DvWUJU0/dqxLL+T5eVIipjsjYSWUO7fV7ZLe/4LyS2FdrtenBW28UUy9r1/aaFrP9Nybrib+vjUOmC8RRDcYOw1HhX5HjJ8anxErE5fdr7BdliU+lTjLm4B4gp1ZEXGZE7Ntx3j/FnF87XsrpYN+tiJ9JFpZoOYHFDW4v+2fWFSUfl9IsBFg9F39ABAOAYRUyghCPpSB8c56+nzkxXIg+d7BwBFjNcG/d+qbAjBgAAkA5g8sXXehtUXBA2ZYvwkwBRnGfYjgssBlTBqs03XAAAAOkAZgD6AFKTOTfjQjHW/Czq3phLPGRB33Qtwo+9XqOmY/js++KC7Fjk4QvqPAAAAOkAZgS1BVIvzFHV8aYmTwoq2oEUSz5k07G1hS8MBgAAAOkA5o54i6Dbuxx3SA28p9fiT4wX95ZQve2Pa6X0NKAoh+n7HQAAAABIx4kEkQbaQmkkI5ZdNMV2Oq0bIiCLEwO3fW6S78AAAAAAIB0nDrI7YboxUxHxcHvn16A3ifM9dw0RDgAAAJAOwAJxpMOy6yMAAAAAgHQAE0EWl/JPsX0QAAAAAOkA5vcwHKdGraBRjwAAAACAdABzJRziQ2royggAAACAdADzBH2K2bSbBQAAAABAOoDZPAD52ekmbX01/Q06AgAAAEA6gJkQjkte4znWPPtq+N+n9O+vUHt0zt2r6KgJAAAAgHQAUxOOLY9v0KfX4++uZID6DgAAAACkA5gaQa+9HhKLff27K2mw+2hyBQAAAIB0AAAAAAAAgHQAAAAAAADSAQAAAADAicX/Ad7y2ZHEqvSCAAAAAElFTkSuQmCC)
w, Add M and N, we get,
![](data:image/png;base64,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)
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)
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LJFQDZwRDZJ51cAZAdDJC9l1yhsSe5wmI8uQKbdpAd8i/7wOQK+ghmBskVANnBoDl7ElklVwBkB4Nl7yVXIDc8soPVsofJFV7QqyM7WCy7Xm6rK3knq+QKgOxgoOyTSK4AyA4Gyj6J5AqA7GCQ7JNMrgDIDobIrmW+7s5cmlRyBUB2MED2bnIFVR+YXMFpM39Hdsi77IeRXAGQHbKR/TfOsxde9tu+7D8iu9Wyk10WuiWqk/LYcXEslF3XMesPrE27eilkz0EVclpVuYbsBZBdCfE6ec7NXNs2BlXISdoExUUDKMpIgIsAgOwAYBH/B+UaRCGQat2dAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Thus, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.
Question 21.Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEsAAABACAYAAABSiYopAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAOISURBVHja7dxPT9RAFADwV5abqBcuCK8e5EA40Q575yC1yr9w2FS22YuHhpC1BorZA8HCbRMkXk00GjjuFxBlP4CRuyaInDRB5RuIYqco2S3bbUm2f2Df4V02m+zML9PXN53XhdXVVaAIF4RAWDFg5QAyANDpE8JlBKhUoMN3zpVKR0Ms2y70MBD2nC/9OhvigWqtDF1GrKKK843mLAjwG5Wi5Yslg7xdsO0euuQcRAUtwkoTFs9t//JAxnvNJxU+uVhIFOvYGYBhTA8rCnuCCLveH0siVkqFGxLAvoNzfBoC/gzKwZFj2QU2BqL4URThEx9UGrCcSS+eSeDD+VepyVkrOhtPA5ZdyvUyHH2ZM4zrznCE1OWstGDxnJSXYdMdB8pbijY3AZDLEFajMVjqkAhw4IzjiNdMbq5CaSdnlvoIyydM07yl38s+cuC+CwB/AKT9XMnuJawmUS4bXSrCWxdMzm8SVlDC51s6AT4DsC9B82x7LPeyVHEBwKmzTFsirKDVZaoSgvyBVlYYrBk2AXj3nVEud6UNazGx51TA96eVjto9IP9MZ/g0zOOmyLH4wPgmtaRJU25dI9/fbPQQLY7gKCIbfTFtWAN8DIZhXMszXMvqRS0VdVZNIVi3F0O1OB/7nlDParVjEYeV5wXL7qfnWRf1eRZhtWHwesx7gyKsZnd0lt8gLMIiLMIiLMIiLMIiLMIiLMIiLMIirBZhVaPEirt3i/9efW9WLhO2QcQ92PDHGhnshu5vI7Y9GMXA4+7d4lCWMT0gIeyc9mWhtHPSIBIM9pBdWQec3EoEK+7eraW52wwF4UfNWcDRSa8DHEszy5OpxqrLBRFjlUrG1TsobKOiL/8/H7StQv9pr0OI4/u2weKnTFm1+MD7udsc0gdVADwMOr5vG6xmkWewAQLba8XK+nqZsfjlqfYJVecu9zpkznqTaJ2VJJaT9GUE9j7MHFtyFNb0XZeAd1+SxOLjnmE311nBHgvz/ZZg+b3r0igaHdsngcVrPHfy52gjSMUha9xYvDjl1XijuyNheVbU0ty4zJi+7AXk7UgXCWsxcihNmkJZe+bNpXwbxJi61qyhLVGsOHu3ONSsxnQEOKx7Z6cmUDUXUruy4uzd4r1ZzW8+wS+bUhfNRUvwhEVYhEVYrcOSquOmifS/DtA5q+DjACzYpf918C9r2v5yO1dhTQiEFUn8BXT2UrdporpqAAAAAElFTkSuQmCC)
Explanation: As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, on adding A and A’ we will get,
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Now, on subtracting A’ from A we will get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Now, Add M and N, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So we see here, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMwAAABACAYAAABWUJhRAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAe3SURBVHja7Z3Nb9NmHMcft9zG24ELkCc7bEgbp9pPekTiUIwrCoiDFZIol02zqqjzVELlQ9c53Cq11a5sHBCdhEYOu/LW3rdx3zRWOG0SAv6DAZ2fx3GaOHbivDhx2u/hi1CU9LGT3+f5vTy/5zG5desWgSAomvAlQBCAgSAAM1bSCZkkhBwKkbSf771KyET4vVcnAAzUJNsunmRE2nEM5L9WpV9p5crUfr7/BZUuBt27RMh7qi6UAQzUAoxClKdF2z6J76MJpDKAgQAMgIGSAgzPd6pVkRNMJin+D8nRJAADjQyYXccADeNaRlXZd5SS50kxtopVPKUQ8sIBZHdP9E2n3AzAQLECYxfZJZJO/5FOkz+5USbF2LjhtyT1Sv4uQjIoETlMpcDmkgKMbemn5fSFO7phHXMuSUIOAwGYNvlUXiab4lqo8kjNli4Tok8CGAjABF1HWZtKE/LKuZZ3fE1F5C5UfqabVgrAQAAmRKZpflJQp7/h8DjgfCBEfqlb9mkAAwGYNlpdNQ5rlDwW0Mj5TZSVIQAT4V6ZRP4mhL0oWtYpAAMBmE4hmkZviHUY05YBDARgOt2vqcmUKL93ul8AAwEYfr85dpnQ2SfG6uphAAMlCZibo7wvd59LdaIxueev5Rhdj7JlAcBAsQLDDZM3NVpZ+apY85i6/tMoN2EVHDDS7MKda0b5c34dhmEczTO6Nl1YyEb5PICBYgWmYaGwqW/LMbrFUdzXQmE623g9aUW9nTPtM5E/v5+AaW3X7twn5M2Ae4reJjEI7frH14PHb73OeK8V+2HGAxjX5ruzA/GPtwBVa3n4wN2/nFu50unDy6UZhRLy1rHHD2LWodrjTonfQA2Td/O6W1/fiZBFub5Jqq3hijdTu++L/1oBzHgAI/riUtq2YVlHegrJbDN3hhGyI4wvpTp/aLXtHyqpdMndC8F2RmkcPFSgEnntJsbzS2HekSfPlGYexn2tACZYfL1m1IWLui3U9/TQtzOlZaUnYISnSZEtSXIb61jOvtxuQCZJO8IjpbStXmZrsZOwOpgEtrZHQ1y3nF2+GtSmIdYLqPYgbi8IYEJshlf7WP5eQrzdTSJJIqLqBuJWYCh9oGnsWwGCEt4fxAeks7NrjFPaIzCDdNH8b2Uymdsu7M6sYVgZAANggmRZxpGLVHo6XSx8ySQnouoiLAsEZsY0zzk5zVPXXdnM/6FVPmCa/Zizc3KSgGGFytyCRhcdaNwOWl/bOYABMLXcm1E6+zO3gwIjd8PsPLqHMW255MSbYV6Gr+xSVlh3z99KFjAimXO+BDdUVLd1wzraLTC1QycORVJASAlgegOm/QGAgzsQMC9P3KNa6Ya4JrdwFKnbui0we12ozUkRd2fiPeXKVFKBaWo7b6icRQWGe6ngQ/haRbXWNREA0xswYQcABn7vPa5F8dxbltgT77fhFVRKyBsekXTqtm4LjKiC1bwMVUtLDe5M8dxZN8AEzdrzKl1yq1r+2aO7veJ+YDyjda7teWPlDCEZQjKR7PNJtGZrhqEfV1NkO+pSSltgal5mx6WvcsoNd+gGK1YuNRhlJGCCZm2RoEvSe//rrGhf6heY+uxRKzfzypkLO4A5qMCIZD8lbQV4LHcdL0JY1hYYLx/wGv8sSz+dSWs/eAaX1JDMD6pbBKBv1az6FXKYg5vDhHVVdxOWdQSmNku/4Qn0xYt0w0uWxgWYpiKAWJDtzRsihxnvHIZ7jpyT7HvRkd/uZyl5EiUs6whMo5chUvOKfgMw2wkA5mbYQmtTEaBHuBGSjXdIJkrJEvst7DepV8t4R3ibxfT6f3gjmmnOUSfIfyRnrauNTWl1l8X2Tjfk77eMmYxMyEvuyuZMk3bbyDYIYLzw6bpCNvcKCK1Fg3oRAMAcOGAsQz8mEntF/UU3jOP+PIU38XJbdvsiyS7T5wuhjbyNM3Cz21Oeez94zfPc9zYJBb+/+4ZGHvoEhTRdQRcQPoUVDQT4VLsPYA4OMLWJ8q+m5/P4IiiviTfK0bf4IQHMgamSDUL4IQEMgAEwiQBmKy5geAzevOFPn+xlsXfYY4uyLoCBWoE5f/YEOfHveds+G4fBlo1rn8mUPKs/m4XKz9yDweOFpt+xv2YfbRB65RGAgYYGjFselV43rlJ3s0t2lGMDGGiowHj7OKhaWKl3W5SLn9bXmPhxrTGFgZHG7rBKDmCgoQLDy+LT2sIX/tdrZX6xf6nTca2jHBvAQEMPycKUZ+SevxMjaWMDGKgdMP8MCxixR4l34bLOz5mMJVSLOLYDzLoDzEMAA/mAGe46jHvcFft1FN6lm7Fx8iXUFzCDaGd3zzb+eKOXPUT9yjtXOerYAAbqC5h+29l506AwQm34R8f2MjaAgUYWkvFFRL5yHlS5it1oehwbwEAjAYbP7sulOYWxworfkOM+2b+fsQEMNHRguMGu5OQrVMl+789zeNsKY9paXNsX+h0bwEBDBYYb7HyWFbyNTkGimnkjLlj6HRvAQEMFhh+63r44kH4V5UlgoxobwEAjS/rHUQAGAjAABgIwAAbAABgAA2AADIABMPsEGHnLPastsDdM2s/3H9YjNz/DD58HMFAgMOLJAUMt/SbIkywO+jEVAAaCAAwEARgIgmr6H+zr6y4jDpRVAAAAAElFTkSuQmCC)
Thus, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.
Question 22.If A, B are symmetric matrices of same order, then AB – BA is a
A. Skew symmetric matrix
B. Symmetric matrix
C. Zero matrix
D. Identity matrix
Answer:Given A and B are symmetric matrix of same order
⇒ A = A’ → eqn1
⇒ B = B’ → eqn2
So, AB – BA = A’B’ – B’A’ (from 1 & 2)
⇒ AB –BA = (BA)’ – (AB)’ (
)
⇒ AB – BA = (-1) ((AB)’ – (BA)’) (taking -1 common)
⇒ AB – BA = -(AB – BA)’ (
)
Here we see that the relation between (AB – BA) and its transpose i.e. (AB – BA)’ is (AB –BA) = -(AB – BA)’, this implies that (AB – BA is a skew symmetric matrix.
Hence proved.
Question 23.If
, and A + A’ = I, if the value of a is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAARCAYAAADUryzEAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAC+SURBVDjLY2hsbGSgBDMMLgPq8yJVjWQZTjEwMPzHjY3vxtbXS2IY0NGRxuMhw7AHv2Y8BjREG/vIekTXp3V08NTneRjKGkf1Ihss65ZTTHQYgAxjMI5aiGyAcXSDD0kGwGysr4+VNGZguEuSATlussXIBhgxyJ7yyKs3JMoAdCeTbADY/wyyb2AaYF4AuQhkeLSxcT3WWICkA2DoMzC8QY4qjOiFBi52A5BswwhUkMGyRqci8+pVB3FeGBADAOfvhSukBCNwAAAAAElFTkSuQmCC)
D. ![](data:image/png;base64,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)
Answer:Given A = ![](data:image/png;base64,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)
Therefore, A’ = ![](data:image/png;base64,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)
Also given that A + A’ = I
(Putting the values in the above equation)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
We know when two matrices are equal only when all their corresponding elements or entries are equal i.e. if A = B, then aij = bij for all i and j.
This implies,
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Find the transpose of each of the following matrices:
Answer:
We know that transpose of a matrix is obtained by interchanging the elements of the rows and columns. In other words, we can say, if
So, let
Therefore, transpose of the given matrix A is denoted by A’
Hence,
The transpose of the given matrix is .
Question 2.
Find the transpose of each of the following matrices:
Answer:
We know that transpose of a matrix is obtained by interchanging the elements of the rows and columns. In other words, we can say, if
So, let
Therefore, transpose of the given matrix B is denoted by B’.
The transpose of the given matrix is .
Question 3.
Find the transpose of each of the following matrices:
Answer:
We know that transpose of a matrix is obtained by interchanging the elements of the rows and columns. In other words, we can say, if
Therefore, the transpose of the given matrix C is denoted by C’.
The transpose of the given matrix is
Question 4.
If and, then verify that
(A + B)’ = A’ + B’,
Answer:
(A+B)’ = A’+B’
Explanation: We will first calculate L.H.S i.e. (A+B)’ and then consecutively we will calculate R.H.S and verify that both are equal.
Therefore,
Now,
So,
From equation 1 & 2 we verify that
(A+B)’ = A’+B’. Hence verified.
Question 5.
If and, then verify that
(A – B)’ = A’– B’
Answer:
We will first calculate L.H.S i.e. (A+B)’ and then consecutively we will calculate R.H.S and verify that both are equal.
From equation 1 & 2 we verify that
(A-B)’ = A’-B’. Hence verified.
Question 6.
If and, then verify that
(A + B). = A’ + B’
Answer:
(A+B)’ = A’+B’
Explanation: Calculate matrix A by taking transpose of A’ and also find the transpose of matrix B and the solve L.H.S and R.H.S separately and then compare the result.
So, A = transpose of A’
B’ = transpose of B
From equation 1 & 2 we verify that
(A+B)’ = A’+B’. Hence verified.
Question 7.
If and, then verify that
(A – B)’ = A’– B’
Answer:
(A-B)’ = A’-B’
Explanation: Calculate matrix A by taking transpose of A’ and also find the transpose of matrix B and the solve L.H.S and R.H.S separately and then compare the result.
From equation 1 & 2 we verify that
(A - B)’ = A’ - B’. Hence verified.
Question 8.
Ifand
, then find (A + 2B)’
Answer:
Explanation: Whenever a constant term is multiplied with a matrix then it implies that every elements in each rows and columns are to be multiplied with that constant term. So in order to solve this question we have to multiplied every element of the matrix B with constant term that is 2.
Now, (A+2B)’ = transpose of A+2B
Question 9.
For the matrices A and B, verify that (AB)’ = B’A’, where
(i)
Answer:
Explanation: The product of two matrices is defined or possible only if the number of columns of the former matrix is equal to number of rows of the latter matrix.
From equation 1 & 2 we verify that
(AB)’ = B’A’. Hence verified.
Question 10.
For the matrices A and B, verify that (AB)’ = B’A’, where
Answer:
Explanation: The product of two matrices is defined or possible only if the number of columns of the former matrix is equal to number of rows of the latter matrix.
From equation 1 & 2 we verify that
(AB)’ = B’A’. Hence verified.
Question 11.
If , then verify that A’ A = I
Answer:
We know A’ can be calculated by taking the transpose of the given matrix A.
Now multiply A and A’. So,
And we know ‘I’ represents an identity matrix
From equation 1 & 2 we can say that
AA’ = I
AA’ = I. Hence verified.
Question 12.
If, then verify that A’ A = I
Answer:
.
And we know ‘I’ represents an identity matrix
From equation 1 & 2 we can say that
AA’ = I
AA’ = I. Hence verified.
Question 13.
Show that the matrix is a symmetric matrix.
Answer:
A matrix is aid to be symmetric only if the transpose of matrix and the matrix itself are equal or same. This means that A = A’.
Now, we know that the transpose of matrix A is
So, from equation 1 & 2 we get
A = A’, hence we can say that Matrix A is a symmetric matrix.
Hence proved.
Question 14.
Show that the matrix is a skew symmetric matrix.
Answer:
A matrix is said to be skew symmetric if the transpose of the matrix is equal to the negative of the matrix. This means that A’ = -A.
Now, we know that the transpose of matrix A is
Now if we carefully look at the equation 2 we can rewrite it as
So, A’ = (-1) × A (from equation 1)
⇒ A’ = -A, hence we can say that Matrix A is a skew symmetric matrix.
Hence proved
Question 15.
For the matrix , verify that
(A + A’) is a symmetric matrix
Answer:
(A + A’) is a symmetric matrix.
On adding them we get,
Explanation: Now to show that the matrix obtained i.e. (A + A’) is symmetric we need to calculate its transpose and prove that the matrix (A + A’) and its transpose are equal. This means that (A + A’) = (A + A’)’.
. → eqn 2
So, from equation 1 & 2 we get,
(A + A’) = (A + A’)’, hence we can say that (A + A’) is a symmetric matrix.
Ans. Hence proved.
Question 16.
For the matrix , verify that
(A – A’) is a skew symmetric matrix
Answer:
(A – A’) is a skew symmetric matrix.
.
subtracting A’ from A, we get,
Explanation: Now to show that the matrix obtained i.e. (A + A’) is skew symmetric we need to calculate its transpose and prove that the matrix (A + A’) is equal to the negative of its transpose are equal. This means that (A + A’) = -(A + A’)’.
We can rewrite above equation as
Also, (A – A’)’ = (-1) × (A – A’) (from equation 1)
(A – A’)’ = -(A – A’), hence we can say that Matrix A is a skew symmetric matrix.
Hence proved.
Question 17.
Find and
, when
Answer:
(i)
On adding A and A’, we get,
Therefore, (A + A’) = 0
(ii)
On adding A and A’, we get,
We can rewrite the above equation as’
Therefore, (A - A’) = 2(A) (from equation 1)
Question 18.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”
Now, on adding A and A’ we will get,
.
.
.
.
on subtracting A’ from A we will get,
Now, Add M and N, we get,
.
us, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.
Ans. Hence proved
Question 19.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”
Now, on adding A and A’ we will get,
.
Now, on subtracting A’ from A we will get,
⇒ N’ = -N
Now, Add M and N, we get,
.
Thus, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.
Question 20.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”
Now, on adding A and A’ we will get,
Now, on subtracting A’ from A we will get,
.
w, Add M and N, we get,
Thus, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.
Question 21.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Answer:
Explanation: As per Theorem 2 “Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.” So in order to prove this we will be using Theorem 1 which states that “For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.”
Now, on adding A and A’ we will get,
Now, on subtracting A’ from A we will get,
Now, Add M and N, we get,
So we see here,
Thus, A is represented as the sum of a symmetric matrix M and a skew symmetric matrix N.
Question 22.
If A, B are symmetric matrices of same order, then AB – BA is a
A. Skew symmetric matrix
B. Symmetric matrix
C. Zero matrix
D. Identity matrix
Answer:
Given A and B are symmetric matrix of same order
⇒ A = A’ → eqn1
⇒ B = B’ → eqn2
So, AB – BA = A’B’ – B’A’ (from 1 & 2)
⇒ AB –BA = (BA)’ – (AB)’ ()
⇒ AB – BA = (-1) ((AB)’ – (BA)’) (taking -1 common)
⇒ AB – BA = -(AB – BA)’ ()
Here we see that the relation between (AB – BA) and its transpose i.e. (AB – BA)’ is (AB –BA) = -(AB – BA)’, this implies that (AB – BA is a skew symmetric matrix.
Hence proved.
Question 23.
If , and A + A’ = I, if the value of a is
A.
B.
C.
D.
Answer:
Given A =
Therefore, A’ =
Also given that A + A’ = I
(Putting the values in the above equation)
We know when two matrices are equal only when all their corresponding elements or entries are equal i.e. if A = B, then aij = bij for all i and j.
This implies,
Exercise 3.4
Question 1.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (1)(3) – (-1)(2) = 5
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 – 2R1
→ ![](data:image/png;base64,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)
Apply row operation- R2→ R2/5
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 + R2
→![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIcAAABzCAMAAACFOuE3AAAAAXNSR0ICQMB9xQAAAKhQTFRFgYGBiIhmZoiIf39dXX9/ampqf25ZWW5/f39AQH9/f25Ibm5ZWW5uSG5/bltISFtugFszM1uAbls1NVtuM0ZuW0g1gEAVbEYdHUZsRkYzRjNGHUhbMzNbMzNIbTMLWjMdHTNaRzMeHjNHMx5HWjMAADNaADRIHR00AB1OMgAySBwAABxIABVAMh0AAB0yHQAyHR0AAB0dMwAAAAAzFAAUHQAAAAAdAAAAmAgEAgAAADd0Uk5TAAwMGRkkJSUqKjExMTFKSkxMWFhkcXJzc319f3+LjY2NmpqaoaGtt7i7vLzHysrK2dnd3eXs7DaGfKIAAALrSURBVHja7Zt7e5MwFMYTmFSZ3SJOUy+TOYsXZutksnz/b+ZT2lVIaM6hEIJ9zvm3b8PvyeU05z2UMQoKa4SF2sTjfOwHy+q5avHE8ePM5zTI0+SIVkqtZwjh+bv7hTuO+Dpg0eoBHkn8vlfuOPj7YANTvkJohUOO3aRMhGOZsQlwnK++Bv45eK6UugkmMB88KZtP8LhPEQd3BA62nAiHvAumwMFz3/sjKtYvGU9vcOc2c5fXPyulvl2i9nL9znG6v/vEcWIcIiMO4hiHQ9UCK6R16cChF1taXKzUej4Ch1FsaV95vGQXqBql97oIC0dYbD6TD2eeOWRlkcTlwi8Hz6vbZ1hkfjnCouLY4bTbSghTSdd15ojLbDst7RukcpkQU6XrBuYIv6CKW1N3xLpkteUxSijcNdnUDbtPq2v4Gj7Rpu6Ic1styCGz5dmbUmFSi67rzrHNHIfzGM8V6kg3dYc4LENtElliyethgcu1DV0bh1FsaZGUdm9SIk1pCeaPflZH/As3VF3n4v4h7oLOOhcc6by7bliO9HbGePr6CN2wHEmp1O3sGB3dk4fkwFmUVFcSB3EQB+UP3xx2hwOwR57a2tZCRh9jz/H8zwukwwHYI/sLrrJYJMYYb3+a8wE6HMLOwfMPAYtyex2jjdG2LqDDAXDEV1sa67Ud5oAdDoHqjC57zgfgcGA5eG4t62AOwOHAcgBVzGgcqd1RxKyLzeHAckhAMtI+FVesLwfgcKA44o+sPwfkcMAcomolR9dBLw7Q4RCA05KUzXfTEGO0/85ZHQ7IHmHJrrdiyafGGHT/IA7iIA6fHPJfW3iBaidTH/n/46jXUR7i03fiII7ROcC2ti5wxAG2tXWBGw6wrW0I3HCAbW1D4IQDbGubAkf7A2xr6wJn5xZsazcF7vIH2NZuCBzmMbCtLd3nD4Zpa9cFDjnAtnZd4JADbGvXBXuOIe9jYFvbFDh5Pxlsa5uC6b2vPZH/NVJMMP4CkLydoflvJVIAAAAASUVORK5CYII=)
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 2.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(1) – (1)(1) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 – R2
→ ![](data:image/png;base64,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)
Apply row operation- R2→ R2 – R1
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFUAAABACAMAAABcMMtAAAAAAXNSR0ICQMB9xQAAAHtQTFRFgYGBiIhmZoiIf39dXX9/f25Zf39AQH9/f25Ibm5ZSG5/SFtugFszM1uAbls1M0ZubEYdHUZsRkYzHUhbMzNbMzNIbTMLWjMdHTNaWjMAADNaAB1OMgAySBwAMh0AAB0yHQAyHR0AAB0dMwAAAAAzFAAUHQAAAAAdAAAARL6qqQAAACh0Uk5TAAwMGRklKioxMTFKTExYZHNzfX9/i42NjaGhuLu8ysrK2dnd3eXs7JlqbfQAAAD5SURBVHja7dhtD4IgEABgKi0qVOw9yspejP//C1tKLpgKTPxQ3n107Bk7Pe4EIYiOYnTj73jOWjo0Z3gs1KPncI/0H9TpIo1tIGl9nUquKbdR5fX1GSBWqrwe1H6oouLLqoe8mqrMUmVaFWdfh69ByOspnNqdqISBCiqooP6eOn5M9GNpVfgnzs9z5eHm0Khqx9iiX/HAStU18EGyHCI/4XfPpYqjwlb+qFqqIvZu9/rJA6vOa+2XZaLii2fZY03UbaDt3MpcaqDS2Hoe0KskQu5VvELuVbLL63Y9tFQbx9gwky4EzFTdGBuK96rUFkyaoJZqJ/dZPY8X4K4sGLre8KEAAAAASUVORK5CYII=)
Question 3.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (1)(7) – (2)(3) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 – 2R1
→ ![](data:image/png;base64,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)
Apply row operation- R1→ R1 - 3R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 4.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(7) – (5)(3) = -1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 –
R1
→ ![](data:image/png;base64,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)
Apply row operation- R1→ R1/2
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 + 3R2
→![](data:image/png;base64,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)
Apply row operation- R2→ -2R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 5.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(4) – (1)(7) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHMAAABACAYAAAAprUjlAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAPkSURBVHja7Z27ThtBFIYndUhHQ8TQkYIOD+kpImtFIF2krC26yAVytqFwgaKBzlLEQ9CFN4gELxCeIJGgTAXPgMPszdZ69j579uIf6ZcorBn7fLNnZo7nH7OLiwsGdUMIAmBCjYf58vcqTghWTYBy8Aj/kfJ4QzD28PLC52XxR8uRuwgurcZ9fqrnwWa8Pz5NhNljvZtjKTcQyOZDBkzAzJ7P65hPmthW0TmxNphnJ5/f9Ti7U/nbFe/d9e3xR4oAOI692++L75yzv7oPVVdbufseWgd+DJ9V/D6cnPXIYUrH2uWMPUUnZSUxPD+sMgDnQ3HItrb+qODHLQTqaKtQ34w/WUPnIACrYipseUQGczodrVmb7JZbQzmaTtc8uPa2xdmNB1Q8UMy7XjDMADDZVhYFO4dof97qNT1+xmCqp1L0na9xkN3RRrCNaTNMD9pynIKMl/Y+SFazA8Gu8GQmKxz0m9ZtkNmW9vpicFUrzPBNpryRVYcZAtPATAJNCtNLEb07qn1qa2EGi0fNoJ9PVcnZrXKYA8Evq17JAiYBzLjGATMhzWpgJqVgEpgqELrVLWC2bAHkblPEUNZRBmv/1mQ5lQYpOG26Mg5TBYCLwWW0PqvKfEJYP5LSxMoXDWL2k+RFA6/09P6LX86b6VR1UNSgkbY4cvt7SUlJReq0InaetqooHKh6rOrTj+ljlkWkMZiq05gvTEm+yPZH9WO0X92Hm79W3Os+V562qlAA0Cu0i9+2I7fLLDiNVoAgGgEmYEKACVW/LdPsRQETMNujLp77XUmYaVsTwIQAEwJMCDABEzABEwJM7DMBE/tMpFmkWWhFYCbdf9A2r2bddzfk7TcDzP2ddbb+b1/KndS5aH7/wUwvGr9JoKwHoXQq6pE0oaLe0G/i9SXjn34ZgemfZos7BzSj8pssLHKeisCM9UgSnMwv4w01ClNZEYIALMo94Mv5NdXNJO7UELq388Es65E0CrUumCoItq2H5T4l3Lqu+szs4qBSpt8s/gx9ai7ukewEzLQAUQZBHcLOarZZyiAlLQKdhkmZYlXA9/jeTwWvCEwTHsmGwzTgzyRKse6c7Q+aQjAN2OrqhFn56TyqFBvtBzANw6RKsS6EiFlpMhm9CYM/mbzNsgE34ZHsLEyqFJt0MaCv5yxF9bYvgDrnnC6bFst6JDsJk7pQYApmWY9kJ2EG81gdIMsuWMp4JE1I9RnAzOMNrQymusiJ8pYRfWZQV70V+wK6qEfSyGKuoDcUxqEOCTABEwJMCDAh0zD9khh+16QZ0rHIBFMwdo/fNWnWExh3PCcRJtSBpxhBAEyogfoPyuAwS6QK7F0AAAAASUVORK5CYII=)
Apply row operation- R2→ R2 –
R1
→ ![](data:image/png;base64,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)
Apply row operation- R2→ 2R2
→![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHsAAABACAYAAAA6eggRAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAPdSURBVHja7Z0/T9tAGMavcxlZqDg2OrAlp+4MlWVR1K1SHYsNMaDUCwMDqky2SFU+BCPfoBJ8gnyCVoKxU/gMoTknDpFzji/2ez7/eZAeCaFwPr+/u/f+2M+FDQYDBrVDCAJgQ42HPft5lyYEq8IQNZktfwnDsz3B2PPsQ9N18YkbhB0EtnrqO/xKzYy9cqd/lQq7y7oPZ2G4hyA2oxEANmDnh21jTDd5vSrOT/LUiRR2GHiHXc7GcmyIxLtj1w9OTN50EHgdxxE/OWd/kzdS5bJz18l3TxYxnsr4fr686ZYOOwzcDmfsJTkhkBL+7amJG7/1xSk7OPgjYagmH1Utu1CdGH+JO5AEL2OuG18S2MPhxY67zx6564cXw+FO3Mvl3+bAxbPJcX8eBDNATJa9jeKVUbIe85m3XnxJYMteLZzgPK0RRK3R4DKtDbDnUNfjGGdUnfoZn433BLtDzy6mZafZdx/jzLm2FyJ6d1ZhLyupURHAzk7hKtibGkKpsOcppjs2vU5vPOx48qvoNG9DZXb2NAq7J/jI1EwcsCsEW1UwYBdM4wrYm1J8KbBlgFSzc8Bu2AQtWoYJPyx/s6ENS6/1VB2neJ3hkhS2DAwXvVHyGerN5bePQri/stIMYGeP29Y3VaQC/9P3xXbpq0qmgiUbUwxEpjLlA/qcD2Z0yraxsSL3w2VdFjGflLpdKi+a8rDc6IsOi9Y+SV5v9YbePiOeth2Ossq2oRhw/CDEC8LDIpNm0h00qDoCbMAGbMCGaq1osplYjwM2YNf3JvGee0tg5116ATYE2BBgQ4ANATYE2BBgY50N2FhnI41DgA3YRLCb5M9Wjf82z5nJe60M2MdHu2z333EYHukWmPQPmz53pUwP9coZM69qmfW0qfzvjtf/ovv/P8T7EeNff5PALuofztVSS/RQL18+TDmcxqSnjcL/Tgabwj9cFIJp2NLOpDpJInqJn/N7U1ks1f/O2cM2GYUMNoV/uMqwZWP2PDXM6B65e2/qvXgq/zsJbCp7StV79qaGbuva2/jfSWBT+YfrCNt0CtfqZJodKQO2/jEbFJbSOsI2ncKz467vfyc7U6WusLPWzVlrWrspfDv/Ow1sIv9w2bA3nPE51bH92EzheRoZ6dFYbZug2Urhef3vZCYBCv9w3WDbSOFF/O/kJxw2eVPFdgov6n8ntf+sbqys+IeN9mpbHuqocc8CXxZoCv87uderiH8439LDjodabmaUcRJUHFMK/zuMfS0SYAM2YAM2BNhQk2BfX3/A93rVSypembAFY0/4Xq/69WDdPX4ErE0ZAEEAbKiB+g9wHCeLaGOcUgAAAABJRU5ErkJggg==)
Apply row operation- R1→ R1 –R2
→![](data:image/png;base64,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)
Apply row operation- R1→
R1
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 6.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(3) – (1)(5) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 –
R1
→ ![](data:image/png;base64,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)
Apply row operation- R1→ R1/2
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 - 5R2
→![](data:image/png;base64,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)
Apply row operation- R2→ 2R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 7.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (3)(2) – (5)(1) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 –
R1
→ ![](data:image/png;base64,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)
Apply row operation- R2→ 3R2
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 - R2
→![](data:image/png;base64,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)
Apply row operation- R1→
R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFcAAABACAYAAABvG+tPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAANbSURBVHja7Zy/btswEMbZucmWJUaoqcmQLTp7z1AoQpt0MxBZ8FZ4EFwBRQYNRsB4M9D6AZKpeYCubQHnBeonSIFm7BQ/Q1yRsh3XluM/tEQyPQO3GIZM/UR+dyQ/ijSbTYKRTSAEhPsM4MafF7MCYc3nNBMuY9VtIOQu/tHDdNB7N2QHCLZJ6g49S2dEHqhTP5sJ1yZ2p8rYNkJcDTrCNR3uc9HrZdqeC9xGUN6zKenGDeuLoHbX8epvTYIaBGXbceCcUvJrEpgyuI3gdcyV9CaEXkAGj52YAPbCh2NiWbeWRW55u7WA22rVNtwdckNdn9VarQ1xzdDbdSnpJIDhziQ9F5B1gctC9wCc8P0s6HE51zOpnNMK7lNRAXKNPTcDuKOeC/4Xk5KaEXC5XFBid02rn42AWwHaBv/iWKc1gEXqcO3hpv2hDmsA89YEtIfLG5dWPaAsrEFnAXwmM51EuGlgPTihUGlP6hqfFgO4n4YTDIS7rM55JW8w/e2nhSr9XSUR8k4i2g2V60VGXKZwQ790+nTiMGPBPSkdyf0iSU+LGdr/EAgX4SJcjLTqIk5+CBfhIlwMhItwES4GwkW4CBcD4SJchItwES7CnQ5dzlusuw1z4B7ub5GtP4eM7Wd1Q2PnLsb21vI36HH3JVDyc7SNQ+1uOWjsyVzzA7xsE/ruhzK4qUaNnD1kybb/4y70I2g5o6BSuFFULYB1dFWLok1VushNgQ4455Pb+4NNyZ6M7Uop3MRGSvqUFr9zG79OBhGj4aZsV4uzEtUoKqgGyx+yePCS8qRcc4MgeOW7pY8D0ImVXxFgDpUfLBGHY4ruV1m5mgM3vzpXGKKHZyUmype8tHfw/6ORxC1YMlarte3+yvhep8sydXZ+Dpk7hYYjScZutRa4sr7X6WupP4gyTGgyD1o734JOln7ZQzHawRVTRup2dLCVilG0496s2hZlcNM0OCmBaDtvSUhrSxTVNsUBRdWau9qQo20Ljq74/J3fGL8Z/l3JD09zX1PgSVSsJQT2eFso+J9le74SuGMZWSQ6q+heeiHbVVKCFa3Lf5IuLX5z/fDNOmQFjXgZajbCRbgIFwPh6gQ3igr4XrHl11bmwo1rwN/4XrEFZnLLvlcMI4MejhAQrpHxF1C00yp8q2zCAAAAAElFTkSuQmCC)
Question 8.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (4)(4) – (5)(3) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- ![](data:image/png;base64,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)
→ ![](data:image/png;base64,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)
Apply row operation- R1→ R1/4
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 - 5R2
→![](data:image/png;base64,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)
Apply row operation- R2→ 4R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix -
→ ![](data:image/png;base64,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)
Question 9.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (3)(7) – (2)(10) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [A : I]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 –
R1
→ ![](data:image/png;base64,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)
Apply row operation- R1→ R1/3
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 - 10R2
→![](data:image/png;base64,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)
Apply row operation- R2→ 3R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 10.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (3)(2) – (-1)(-4) = 2
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 +
R1
→ ![](data:image/png;base64,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)
Apply row operation- R1→ R1/3
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 +
R2
→![](data:image/png;base64,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)
Apply row operation- R2→
R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 11.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(-2) – (-6)(1) = 2
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 –
R1
→ ![](data:image/png;base64,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)
Apply row operation- R1→ R1/2
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 + 3R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 12.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (6)(1) – (-2)(-3) = 0
So the matrix is not invertible.
∴ the inverse of the given matrix does not exist.
Question 13.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(2) – (-1)(-3) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 +
R1
→ ![](data:image/png;base64,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)
Apply row operation- R1→ R1/2
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 + R2
→![](data:image/png;base64,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)
Apply row operation- R2→ 2R2
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 14.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(2) – (1)(4) = 0
So the matrix is not invertible.
∴ the inverse of the given matrix does not exist.
Question 15.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = [ (2) {2×2- 3×(-2)} – (-3) {2 × 2 - 3×3} + (3) {2× (-2) – 2×3}]
= [2 {4-(-6)} + 3 {4-9} + 3 { -4-6}]
= [2(10) + 3(-5) + 3(-10)]
= [20-15-30]
= -25
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→![](data:image/png;base64,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)
Apply row operation- R1→
R1
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 -2R1
→![](data:image/png;base64,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)
Apply row operation- R3→ R3 - 3R1
→
.
Apply row operation- R2→
R2
Apply row operation- R1→ R1 +
R2
→![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOEAAACtCAYAAABGKRhtAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAA33SURBVHja7Z0/cxTHFsV7lT7zIggM9EbGVJF5W8oVuJYRgpdtwWpKkasUTOltWaCqCXBp5YyywTlFAnwAh36UITfSF7BdFqETSV8B+W33zKx2Z2eGHe3M9L0zx1Un4I/h0t2/6b7d53aL77//XkAQZE9oBAiiBOHov6U0obEgKAdYQrQyeGolQjgcbn6uhPgw+g0fZyWPncHwKzQuBM2ngSMfJrMkzmR3+1EqhB3Rebs5HH6ORoSgcrTdlY8AIQQBQohQvtJiEm9t9icAIeBreV6v0+2qPSnFn/HBQFGPvd6XSor345xKdg57nn8TEEIstb+p1kW7/Xu7Lf4YDeh/qEP42Pu6s7zs/LD15MlnZowO+jdGQB4IoT5s+v5VQAjxhdFVd6lD+OTJ1meOcr6LAByP04HzlRTiVLn7dwEhBAgtzY4jCE8AIQQI7eS0S30lXgnlvkROCAHCauHTG0pqNAUeCuX8vOX7lwAhBAgrzA3XpPg1cpuYuDsbP3EFERBCbJejGsaBu3J/lA8eh7HvAkIIEFpQuDt6Yo4pGI5VQAixh1DLVeIlIIQAoeWBLKTzNn6GCAghQFiwkryivr91yZHirXQGD5ETQiylB/Wwr+5pCEVn4zVVI7exqAlxFHhFPaXj1AC6Sj6Vyn1quQ1bFzWUA0KLX3MKFQDhpsZxvNCU4qxibGvL7edTccrlXxx3cMc2gNv97nrgYdWGcnXwtfdYzfshA4QVq24VANCIE7OKkCfRx2DHddaMja4/vAcIyQGYUQGANmYp39+8qq+AiUMUXGUxX2UHIKxyKVXDCoCmy+zK6lkwdt9S1K/zGAg0sIDQ8uxoli6b++toD4Yf1uvinbjuvJv5uAaXpB0JtfHqU3+O2ZmO/T5AWF1Cv7TBvAKg0blgdBthAoRZgAJCGvDVpgKg0RCGS86k2W4M4Rz5PiC0sISpUwUAIASErGG0VQGQdas6lZvW88Zo49+RlfdlLVUBIc2vaaUVACm3QKfKEoQfi1YpGzNSvM3YmPkwT74PCAnIbM7grJCl0s4D8xw9AUIC2u7KXa4VAFjJBCsZnU5ENjVjY9N9KtTRPOwAQgt5zuTPmQqAUQLPtQIAimZDeRL6RZe2+yt9nevPa8AoHELb16kv4mYv94tJtwIAKgDE8002Y+DuD4Y35v1/C4OQwnXqKW72DoWynPQKgJ01DOJmqzAIKVynfu5mDwZ2Xjc7BLGGcOoPtADh2M0ey63yuNmRr9J86WieFIdqGtIoCIPdKHma4WZneXdKlR8wU1k/1mxlgA345klxFi2qBYRF5VvZbvYP87jZm6qgHCd2uE3AVD5virNoUS0gLCIXLMjN3thZsH37hfauUp05ssZUEUW1gLAICAsy0jZRGx3x2vSXVG+6/e11ivlU1pgqoqgWEAJCa4pd8mQqOsw5pu9f4wBhUUW1gLDI5eiCbvaSNhesu/0/Jc/zvnCdlW9DIM+oLePSxlRd0pD6bMwU4GYvCULrbv/c7RjeP0oewpqsgGpzRJGWiId3uOAipXyriiNKgxcQMoFw0s0+MQvlcrNDkx+02TNXosvRI4ppiDUIbV+nPuFm70y52XGT2QU+aJ1D6jMh5TTECoRUrlOPudnf53GzQxODglB9Y9bqqoii2totRyE+Cv2WrfjPbSj5jMpS9FMQFlFUCwgha9KwtdXtF/otjMn6xhV3cJ/Qh2IixXmQmOIsWlQLCJs5A8XPBls24phauo/UXnaeU1q+50lxOKchgLBi+V7vprn0N6pWkJ3D0C7WQvs0OAcHhNUoenci9mU3djEUHgNCQFiyou106bh7k0+jhQ6VM3hbASEgrCC/WXEG3yTCaZwddA7HIUDYOOHyX0AICG0vU/VMiOfRACEgtKNgs0YdoI0BISC0oMidAm8rIASElgDUVzPg+nsIEFps+KTdUggQAsIKZkCdByrl7iX9GtoIEALCkgHUZmT9PHbcP6rtbEo5P+JaRkAICMvMAQN3/+n0TdfnonpLuO2XtrjFBQiJSrv8sy94ksfUHDMUXtriFFdJEK7euiwu/706HN4CSA1cJhF4aYtTXBf+96xe2RFXVn/DTAhlf6kJDnaqcWE5CgFCQAgBQkAICCFACAghQAgIASEECAEhBAgBISCEACEghAAhIASENVcVD4sygJD8k9iAsKY6v6Fa/VVWH9l+aWu+uB6Qics6hBOudivPPsdiINshFNopH+R2X9riENcifVoYhEGpTnddSXEQvgdwEL4T2KqqETg461PaSaGgl3VKsFCfFvc+oVkayBPH3VnTP95xnTV95XtV17tzcdaft9Pgjo12guj1aSEQ+v7mVf0qanw5kPaAYxM3EabaKRabjXaC6PRpIRAGDzLOXuMevZZaJRCUIaTUTlBhffrIzILpfbpbOoTjG6Qz3w2f/guaCGFZ7UR9c6fOmqNPj+bp04UhHA+ghECygmwahGW0UxVHFFD5fbo4hOG0m0T8+YtD1T12QhZCYu0E0elTQFhTCOPnVp9QC1BxhjAjn8marhu7HK2onYLduazb3ZIPuXOAW2kuSjGurLwvT58WszGjX5vN3HCo7tkv0hszhNopY7B/zKMmx1VUnxZyRJF2JhK+0X6q3P27TYcwq52iZU2V7QTR6dNCIAz/wpPJM5Hw1aHdUYBHVW42UIaQUjtBxfdplFvn7dPCbGvBF0GehH7RpfDK9+Mq396j6vhPaSdVRDvhnJBwn865sim0ikJf9T52t0v1vj8Y3qj4i0TO8V9mO+GckGyfHuTpU9QTQpBlAUIIAoQQddk8H+QYFyCEClVUqjP9luJs1QDiAoRQSQpKdWKH4ar/EnEBQqiq2aZ9+8WW71+idNQTxdXb2vp3HXyvgBBK1UZHvDbGB6nedPvb61Ryrg0lXo3juu/d5X5OCgihRMXOXc9MziU7hz3fv4a4ACFUoTzP+8J1Vr4NB/4ZlbtwqMYFCKHSNK4YCC2BlOJak+JXanEBQqicpWBUP0fsBgCqcQFCqBSFZuVTaudxVOMChFBJGyOdQ2pjgmpcgBAqZ8BI521V15VwjwsQQhdWeFFUK/5zG0o+s7nkS4vLVfIpx6UoIIRSpWFrq9svep5/Ux+G+/7WJT3QV9zBfZtx6RgoxgUIoeI3OiaLVEdqLzvPqyzS5hYXIJxdsrAucalLiQ7UUAi5l7jUqUQHaiiEySUu9u/zbEr8UMMhpFp605T4IUBItvSmKfFDDYeQe4lLHUt0oIbmhNxLXOpUogM1FMJIU6U3Fb4UXEr8TEt0oIZDGMXPucSFe/wQIDTiXOJSh/ghQMi6xKUO8UOAUJhXmpiWuIw7iHH8UIMgpFp6wz1+eFiZQRgOJCsdRrX0hnP8j73el0qK92P7nD63HMUHcIhCaF4n7XfXR512EL3RFj4Y2qpkA4N5iQu1+PVT58vLzg/RMng46N8I+hbnlmQhDF7IlSeOu7Omf7zjOmv6GWHVH95DQ/OSOaNUznfxPDTPO+xQxRBG5TfxV3GD7XV8OesiPTuaDysgpAfhdlfuJp1lRV9O2d1+hMbmLZ2nmjcgUFZVNYSrty6Ly3+vDoe3Mpcu18U7cd15N7N8CRwfHzjaxqDzXN/zemo0BR4K5fysy6zQLgVDuHplR1xZ/e3CM+EYtAQIswCFeOSG4RXz48oO2dn4CSASW45GS86k2W4MIbyP7GGc2L39Z5Re7KJdACG5JVvS5UwpanH9d4Z9fYKPKjUIM/K+rKVqUZsFeVVGIwa7wLF7YVIU30GmEH8eBQ90AkJSEI7r3jI3ZsrZUZt34E+K4CzKKn6zEw4vKy0Iz2eC2fPA8FwJh7s8l9gzM6+20un0Ij6bQwQgjHKFyYTd2NjM+aE6wtKFW+7Xv2GKiY1X1FOTXlap3KcU83BiMbUqh/B8NpQnoV90abu/0te7aWpzfx0Dm5dMirHcfj6Vy8rlXyJLIoXBnuBTVrY2vcKz1E63q/akFH/mNacUWkUxZUKW6j33NwIgojP12Kc8uKN/bNunvL+p1kW7/Xu7Lf4Ij3DsQQhBZWvsU44NdAo+ZQMTIITqruCZgNm3OSZ8yruA0H6+wPpwvE6H+6Xkqtk+5SObPmVAqJcqXu+mMRpHrxnJzmF4nTyLgcw9/tJzQeI+5cZDGNW6xQ63jemYQ2Ex9/grgZC4RbLREEauHem4e5PXMYQ3WJ9Rt1kVHX9dvayAkHjnrDiDb9I7hvbluUXHX1cvq02fMiBcQNwNx1XHT9nLatOnDAgX6TQ9kzC9koF7/GUo7TyQwiVUJUH46est6G92qAOuHxHu8ZeYFxqfcpTPUvEpT0CY66xyf/XKw4zrLfhCGN1ezdW7yj3+8mdD41NWUz5li7OgjiOw0+mn7B68zrPptfAdM1QHsHZWcC254R5/JSAS8inHXllO3fRqVE6o/1FJu42IH6La37XJCfUMovMopdy9pF9D/BBF/Vf965mQ/3nDHkI9SPWaXF/JFz/L0nYwpZwfKV/JQDV+eFkB4fw5VJCcn459lzFRzq/mit/CLebwsgLCXEl69qGyPKbsmKEYP7ysgBCyKO5eXEAIsRd3Ly4ghGqt0MuKG/QAIWRtmQovKyCE7AleVkAIWRS8rIAQsgwgvKyAELIoeFkBIWRxBoSXFRBCFgHk7MUFhBD/HJCglxUQorEaI+5eXEAIQdBFIPxfCoR4EAaCypY5/okt7wEhBFGDsOf712zfvgxB3JX2PIG+rjETQiXEX0jOIWhxZT1PMPMcARoMgizPmmgECLKr/wMI2ynqsDuhdwAAAABJRU5ErkJggg==)
Apply row operation- R3→ R3 -
R2
→![](data:image/png;base64,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)
Apply row operation- R3→
R3
→![](data:image/png;base64,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)
Apply row operation- R1→ R1
R3
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 16.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = [ (1) {0- 5 × (-5)} – (3) {(-3) × 0 – (-5) × 2} + (-2) {5 × (-3) – 2 × 0}]
= [1 {25} - 3 {0 + 10} - 2 {-15}]
= [1(25) - 3(10) – 2 (-15)]
= [25-30+30]
= 25
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [A : I]
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 + 3R1
→![](data:image/png;base64,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)
Apply row operation- R3→ R3 -2R1
→![](data:image/png;base64,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)
Apply row operation- R2→
R2
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 – 3R2
→![](data:image/png;base64,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)
Apply row operation- R3→ R3 + R2
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 -
R3
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 +
R3
→![](data:image/png;base64,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)
Apply row operation- R3→
R3
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 17.Using elementary transformations, find the inverse of each of the matrices.
![](data:image/png;base64,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)
Answer:First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here |A| = [(2) {1×3 – 1 × (0)} – (0) {3 × 5 – 0 × 0} + (-1) {5 × 1 – 1 × 0}]
= [2{3} - 0 {15} - 1 {5}]
= [6-0-5]
= 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [A : I]
→![](data:image/png;base64,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)
Apply row operation- R1→
R1
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 -5R1
→![](data:image/png;base64,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)
Apply row operation- R3→ R3 – R2
→![](data:image/png;base64,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)
Apply row operation- R1→ R1 + R3
→![](data:image/png;base64,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)
Apply row operation- R2→ R2 –5R3
→![](data:image/png;base64,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)
Apply row operation- R3 → 2R3
→![](data:image/png;base64,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)
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→ ![](data:image/png;base64,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)
Question 18.Matrices A and B will be inverse of each other only if
A. AB = BA
B. AB = BA = 0
C. AB = 0, BA = I
D. AB = BA = I
Answer:Here it is given that A & B are inverse of each other.
∴ A-1 = B -(i)
Also B-1 = A -(ii)
From definition of inverse matrix, we know that-
→ AA-1 = I
∵ A-1 = B {from eq(i)}
→ AB = I -(iii)
Similarly, BB-1 = I
∵ B-1 = A {from eq(ii)}
→ BA = I -(iv)
So AB = BA = I {from eq(iii) and eq(iv)}
Hence option D is the correct answer.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (1)(3) – (-1)(2) = 5
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 – 2R1
→
Apply row operation- R2→ R2/5
→
Apply row operation- R1→ R1 + R2
→
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→
Question 2.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(1) – (1)(1) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R1→ R1 – R2
→
Apply row operation- R2→ R2 – R1
→
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→
Question 3.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (1)(7) – (2)(3) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 – 2R1
→
Apply row operation- R1→ R1 - 3R2
→
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→
Question 4.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(7) – (5)(3) = -1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 – R1
→
Apply row operation- R1→ R1/2
→
Apply row operation- R1→ R1 + 3R2
→
Apply row operation- R2→ -2R2
→
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→
Question 5.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(4) – (1)(7) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 – R1
→
Apply row operation- R2→ 2R2
→
Apply row operation- R1→ R1 –R2
→
Apply row operation- R1→ R1
→
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→
Question 6.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(3) – (1)(5) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 – R1
→
Apply row operation- R1→ R1/2
→
Apply row operation- R1→ R1 - 5R2
→
Apply row operation- R2→ 2R2
→
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→
Question 7.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (3)(2) – (5)(1) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 –R1
→
Apply row operation- R2→ 3R2
→
Apply row operation- R1→ R1 - R2
→
Apply row operation- R1→ R2
→
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→
Question 8.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (4)(4) – (5)(3) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation-
→
Apply row operation- R1→ R1/4
→
Apply row operation- R1→ R1 - 5R2
→
Apply row operation- R2→ 4R2
→
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix -
→
Question 9.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (3)(7) – (2)(10) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [A : I]
→
Apply row operation- R2→ R2 – R1
→
Apply row operation- R1→ R1/3
→
Apply row operation- R1→ R1 - 10R2
→
Apply row operation- R2→ 3R2
→
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→
Question 10.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (3)(2) – (-1)(-4) = 2
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 + R1
→
Apply row operation- R1→ R1/3
→
Apply row operation- R1→ R1 + R2
→
Apply row operation- R2→ R2
→
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→
Question 11.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(-2) – (-6)(1) = 2
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 – R1
→
Apply row operation- R1→ R1/2
→
Apply row operation- R1→ R1 + 3R2
→
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→
Question 12.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (6)(1) – (-2)(-3) = 0
So the matrix is not invertible.
∴ the inverse of the given matrix does not exist.
Question 13.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(2) – (-1)(-3) = 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R2→ R2 + R1
→
Apply row operation- R1→ R1/2
→
Apply row operation- R1→ R1 + R2
→
Apply row operation- R2→ 2R2
→
The matrix so obtained is of the form –
→ [ I : A-1 ]
Hence inverse of the given matrix-
→
Question 14.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = (2)(2) – (1)(4) = 0
So the matrix is not invertible.
∴ the inverse of the given matrix does not exist.
Question 15.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = [ (2) {2×2- 3×(-2)} – (-3) {2 × 2 - 3×3} + (3) {2× (-2) – 2×3}]
= [2 {4-(-6)} + 3 {4-9} + 3 { -4-6}]
= [2(10) + 3(-5) + 3(-10)]
= [20-15-30]
= -25
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [ A : I ]
→
Apply row operation- R1→ R1
→
Apply row operation- R2→ R2 -2R1
→
Apply row operation- R3→ R3 - 3R1
→.
Apply row operation- R2→ R2
Apply row operation- R1→ R1 + R2
→
Apply row operation- R3→ R3 - R2
→
Apply row operation- R3→ R3
→
Apply row operation- R1→ R1 R3
→
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→
Question 16.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here ∣A∣ = [ (1) {0- 5 × (-5)} – (3) {(-3) × 0 – (-5) × 2} + (-2) {5 × (-3) – 2 × 0}]
= [1 {25} - 3 {0 + 10} - 2 {-15}]
= [1(25) - 3(10) – 2 (-15)]
= [25-30+30]
= 25
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [A : I]
→
Apply row operation- R2→ R2 + 3R1
→
Apply row operation- R3→ R3 -2R1
→
Apply row operation- R2→ R2
→
Apply row operation- R1→ R1 – 3R2
→
Apply row operation- R3→ R3 + R2
→
Apply row operation- R1→ R1 - R3
→
Apply row operation- R2→ R2 + R3
→
Apply row operation- R3→ R3
→
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→
Question 17.
Using elementary transformations, find the inverse of each of the matrices.
Answer:
First of all we need to check whether the matrix is invertible or not. For that-
For the inverse of a matrix A to exist,
Determinant of A ≠ 0
Here |A| = [(2) {1×3 – 1 × (0)} – (0) {3 × 5 – 0 × 0} + (-1) {5 × 1 – 1 × 0}]
= [2{3} - 0 {15} - 1 {5}]
= [6-0-5]
= 1
So the matrix is invertible.
Now to find the inverse of the matrix,
We know AA-1 = I
Let’s make augmented matrix-
→ [A : I]
→
Apply row operation- R1→ R1
→
Apply row operation- R2→ R2 -5R1
→
Apply row operation- R3→ R3 – R2
→
Apply row operation- R1→ R1 + R3
→
Apply row operation- R2→ R2 –5R3
→
Apply row operation- R3 → 2R3
→
The matrix so obtained is of the form –
→ [I : A-1]
Hence inverse of the given matrix-
→
Question 18.
Matrices A and B will be inverse of each other only if
A. AB = BA
B. AB = BA = 0
C. AB = 0, BA = I
D. AB = BA = I
Answer:
Here it is given that A & B are inverse of each other.
∴ A-1 = B -(i)
Also B-1 = A -(ii)
From definition of inverse matrix, we know that-
→ AA-1 = I
∵ A-1 = B {from eq(i)}
→ AB = I -(iii)
Similarly, BB-1 = I
∵ B-1 = A {from eq(ii)}
→ BA = I -(iv)
So AB = BA = I {from eq(iii) and eq(iv)}
Hence option D is the correct answer.
Miscellaneous Exercise
Question 1.Let, show that (aI + bA)n = anI + nan-1bA, where I is the identity matrix of order 2 and nϵN.
Answer:To prove: (aI + bA)n = anI + nan-1bA
Proof: Given A ![](data:image/png;base64,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)
We will be proving the above equation using mathematical induction.
Steps involved in mathematical induction
are-
1. Prove the equation for n=1
2. Assume the equation to be true for n=k, where kϵN
3. Finally prove the equation for n=k+1
I is the identity matrix of order 2,
i.e. I ![](data:image/png;base64,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)
Let P(n): (aI + bA)n = anI + nan-1bA, n ϵ N
For n=1,
L.H.S: (aI + bA)1 = aI + bA
R.H.S: a1I + 1a1-1bA= aI + a0bA= aI + bA
So, L.H.S = R.H.S
∴ P(n) is true for n=1
Now assuming P(n) to be true for n=k, where k ϵ N
P(k) : (aI + bA)k = akI + kak-1bA …… (1)
Now proving for n=k+1, i.e. P(k+1) is also true
L.H.S = (aI + bA)k+1
= (aI + bA)k . (aI + bA)1
= (akI + kak-1bA). (aI + bA) ……from (1)
= aI(akI + kak-1bA) + bA(akI + kak-1bA)
= aI(akI) + aI(kak-1bA) + bA(akI) + bA(kak-1bA)
= (a.ak) (I×I) + kb(a.ak-1)(IA) + (bak)(AI) + (bb) kak-1(AA)
= ak+1 I2 + ka1+k-1 bA + bakA + b2kak-1A2 (IA = AI= A & I2 = I)
= ak+1 I + kak bA + bakA + b2kak-1A2
Calculating A2
A2 = A.A
![](data:image/png;base64,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)
∴ A2 = O (O is the null matrix)
Putting value of A2 in L.H.S
L.H.S = ak+1 I + kak bA + bakA + b2kak-1(O)
= ak+1 I + kak bA + bakA + 0
= ak+1 I + kak bA + bakA
= ak+1 I + (k+1)ak bA
Putting n=k+1 in R.H.S
R.H.S = ak+1 I + (k+1)ak bA
∴ L.H.S = R.H.S
All conditions are proved. Hence P(k+1) is true.
∴ By mathematical induction we have proved that P(n) is true for all n ϵ N.
Thus,(aI + bA)n = anI + nan-1bA, where I is the identity matrix of order 2 and nϵN.
Question 2.If
, prove that
.
Answer:We will be proving the above equation by putting different values of n (i.e. n = 1, 2, 3 ….n)
For n=1,
![](data:image/png;base64,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)
For n=2,
A2 = A.A
![](data:image/png;base64,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)
For n=3,
A3 = A2.A
![](data:image/png;base64,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)
![](data:image/png;base64,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)
r n=4,
A4 = A3.A
![](data:image/png;base64,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)
![](data:image/png;base64,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)
d so on for other values of n.
If we notice each result, then we will see that it is of same type that we are trying to prove.
So we can generalise the above results for all n ϵ N
Hence Proved
Question 3.If
then prove that
where n is any positive integer.
Answer:We will be proving the above equation by putting different values of n. Because n is a positive integer so it will take values which are greater than 0 i.e. n = 1, 2, 3 …n
For n=1,
L.H.S = An = A1 = A = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEwAAABACAYAAACwVZFQAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAKISURBVHja7dw9T8JAGAfwx1ncXDQemwsbPdyZamOMG4nQuDIQ7cLARA42EsMXcGNzcdZEPgGfQBMcnfQzqL3jRUGsbdOXK/xJLmEg9Llf77ne0x5Qt9slNP8NCABLEMx9bf3V1hohQJ/nb4S42ONEL+4HP3439mY5oriOWJcma67uM30y87LpCWaQ8XghxB5SbwIJsLTBsji/+Y03UjDhVA8NRqN5vjNjVHXEYTbmK/7ip4+RgQnHKnJuXdd7vdwinr9A0moybkb0nihYr1fPmdxsz7CWg+F251RLLNm3SUZ8Jj7CvM6ermA1zvrMsoV1QEMtwGqcBsRrAx2xOjY/ZbzWl9mROlijUTHUUC9Zd8tpqksqlljpVvYpVbDvg09WxGpV7J5F3dBkKs4qFC1GmAzCsY/OJ1ef1aVE3PXeX/Xfcoe1AAt7yY6w3vvVZCdVPO6I/wnZatV35mCt1v5/C9jYSyM18WuyFvMqqKfNfc/HXrHGDqaCPLCGOk7+qabkqqE8CybOOSyTYLIMUvfM3Nqx0mgYEk8GohaG7pyhcx2ZCpg86DHP3yxMsqx0b9nOie6F9w+wcepXSdwPAxjAAJZA0b7qhgLAAAYwgAEMYAADGMAABjCAAQxgANMdLEv7w4LE6gOsXNil3deyEAU/X+g41aJp8jZj9Kzzw4+wsV7x7T6xs4dIwJR+Pv8kA4j7iXckqRUi1kjBFoLRHCxsrAADGMCyBhZ8WbHOYLE8NUoaLOgesY0GC7pHDCMMKQkwgP03Ac+CkEWqzkV4mFgj/nGW2gT85meyTbuFjRWbUUJclQEGMIABbP3Bpj8r2fT/rfAFxonG+N8K76XHxqde4NEIBIDF2r4At+fJH/nkxBAAAAAASUVORK5CYII=)
R.H.S = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ L.H.S = R.H.S
For n=2,
L.H.S = A2
= A.A
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ L.H.S = R.H.S
For n = 3,
L.H.S = A3
= A2.A
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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKUAAACDCAYAAADs3tvxAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAf3SURBVHja7Z0/b9tGGMavc+MtSwyfpsZDtujoMUCGQiHapJuBSIS3QoPgCmhigIPrMt6MNu7uTPEUZMiaFrC/gP0JUsAeO9mfIaruSP2jjxLJO5J35BPgATJYNu/en+59799Dcnh4SCDIJKETIEAJQaVCOf73TZLQ2Q0HLQMX2v5oEOw8YIRcj//Q17uiN+4weIzgNFNDl76Sc0G+0s7u60KhbJP22U4QPEAgoDTa7dDXxkFZZlpXLSfqVo5kbUMR7TcKysFgu93psANKyb+yh9Kt/cH2ZpuSy3FnjoRo+7LT3f0xdduG3YeMkotp2hl/fnuwv2nzCEUIu047gBTVfmOgfOOx56TV+tJqkS8ckKKh3B98P+aR3MZqGQEn6wYvVgfEfcyY+2f/6OjeYoDSB9Ukzfoj3fPz9juO+0cR7TcufQs4C4by6Kh/z90g59T1gvlOdSk5C8Fc3rH88x3WOZh8dj5QPLDMe/PcJiBFjKYZYzVUov+Y+1tR7a8NlEkNSRzlOsOfk2AlhN7mWRWwFcoeo8f8Cxq2Pf9IBygVoFweIHKaJzC8yBefZd57m4Dk/U1Z73j2hcwHpc72A0rZSJmhY3kw+ARNpD/H/dT3/TWb0rZDnQ88NnmhLKL9gPJO+mlfpn1WEciwDp1OlGi791e81jI5bU/KlDxQJrZfEUxroYyvj40bsjf+zJ7KuhkPUp56iAdn6G29HNdTN2WsHKTtk2WKB14lfUvav9c4KMP1NPn2VFxpIdM30qZfVtGpZVt2cbWePHnH68h5SH2/vzaF0vfX8yyE62o/0nf0N2Wz8TInSoYAPIo0/j+7ytMGHe1vPJThIrgXyFJh3g4lG+65LXWljvStu/2NhjLoshfxNMbFtx/nd2tkgMpq1kkKrKqmLBLKeFtl7Z9uSii23ygoeSM5KCKFsN5pltEqK5S73a1utM04kmnyu6L6deFLIrbU+HE8sdc7aE9qMrEIzby3tgG5Csp4HxTdfmOgjIrkmzTn6lSh5DPF5ZOB2TnPyaTqzkzVaZ0sPCd1Prve8AcbgVxc3rlbS05q0EkfFN1+I4+uQc0WoIQAJQQBSghQAkpIVWJZkPVOASUEKCFICcq0J08AJVQKlFlO5MiKU0AJIX1DgBJQQqgpIUCJmhJC+oYAJaCEACUEAUoIUAJKCFCmU53MTetuKlsClE8f3Sf3/3saBI+qCmJdzE2bYir7C/v2mNCf/qktlHUxN63IVPZGxVQ2r6lqraGsm7npFE4LTGVVTFVrP1ImfYsB5YrsUqGpbOOgtNXctEwol6kMU9nGQGmzuakpUJZlKlsClNUvCUnNPVnv2DYTqqqhDNMvuyjaVLVRtxlNMTctA0qV44bJqVubqeweoEwouG18500aKLMYqBbh1aTa7429922Duakp6btsU9nGQmmruWnZUEamsr/LyoOi+r32UNbN3LRMKHWbyqY1Va01lHUzN1Uxlc08olVoKltrKOtkbqpqKptFWUxl44aqOvodBleQcQKUEKCEIEAJAUpACdkDZfS+vzrdlYHUleZlpoVAyQi5WrW0ADVPy/boC4USgrSNqugECFBCEKCEACUEAUoIUEIQoIQAJQQBSggqEEpd944b0fHop+KhjPa+r7H3vVpZ94IBpQKUOLqmJh1GAIBSAUqdNiNlp90s7QKUlkA5l+pHcpnnaJHG8lnVphlQVghldNE+6WrnSGbOXqXSWD6r2jQDyoqh5JfVZXeDQ0s5+tHUSVGSa4WqTTOgrBhK/nPdrhw6cRGfuh9N9f1JglKXTTOgNHD2bXpQ8vj7ZHF9A5SGQWl66s4DZVabZkBpGJSmp+48UGa1aQaUhkFZVkBU7JizQpnVphlQGgRlWak7srPLbcecBco8gAFKg6C0IXVngTKvTTOgNAhKW4KRBkoVm2ZAaQiUNsy600KZ1qYZUBoOpUjd40Aa31ErLJ/T2jQDSgug9Bh5b/oLPVdZPmexaQaUFi0JQYASUAJKQAkBygqgNO8l9LYp6VXDgBIjZWUSS06GHXAGlIASUAJKQGk8lConaAAloNQOpeoJGkAJKJG+ASWgBJSAEjUloASUqCkBJdI3BCgBJaAElBCgBJSAElBqgLIO1so6vTUBpXYosx1d465kjJKL6ayeti+3B/ubNnWe3GcznZmVTDi6ViGU4TXU2e2+GaDmmaUuk3QZLaVvEKA0CEp+pbbDOgfx66bR5axb0y+RTeT7O+us9exd3/fXdP1OQFlx+paNnjZBGVr8kRGlzt/cQlpHTQwoDYKSB1QEWSH1lVpL3r1yK/zNd3x/HVBaDiWHkRvcC/N6x/2kMxWWocFg8J3nbv0aARraSCuACSgrXhIK7VqEJ/h0tOFOGaabXK1oi9JLBHCbsWIo5wPKHSYmo01VQVE5ITVpPyPkSmUFAVAaAmV8olPFspDqCanF35N/nRJQGgblbEZr11rl3S/WeMKDkbI+UIqRZsM9t7Gu5BLbhNQ9y/v8gLJCKGX1me/318QLkiwIiuz5w2UteqziswkoK4JSbCny/WKx1z1o82ByIHlAKfPe2tBp/Flb7Nk7vlc///xb3vClam0LKCuAUiyfOK2ThckDdT7LXotnquZWC8Tztxz3pDsMHuqYcAFKnKc0SoASUAJKQAkBSkAJKAEloASUpkLp++s6763UtuMlfQQoC4AyPJCQ/5UdTdHQpa/y7K0DSggClBAEKCFD9T8Hm2qcnaYhGQAAAABJRU5ErkJggg==)
∴ L.H.S = R.H.S
For n = 4,
L.H.S = A4
= A3.A
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALIAAABACAYAAAC+0C16AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAVOSURBVHja7Z29TtxAEMc3daCjScReFSjouD16qsNKIB0SxqKLKE7kpITiCoQMHVLEA4QKqogibRIJXgCeIJGgTHU8Q4jX94m954+175jd+yONhE6A1//9eXZmdhmz4+NjBoOZbhABBpBhMCtBDr5ejDKInU0r6KqnU2kX9P2dV4Kx++Ai/+LG207TXwbAHdur8321TuyR1/f2oWvHmg7/nFWnUkGusurVju+/Aqz6gKtAhq7pOgFkgAyQiwieFPdRjv2yjE33XsoA2dQYuqhOzwLyUNz3qDZxT80DNRqb1XpdHHLO/kRFHLaDxuZilbPb/r3w6m3d3Xs3CZD9prsgOLvpx5LBtTcbB4tm5AzZ55wMyEeeWB8RxIeBPBPbF5SEDsdbqfyWEKsSjQFIzjJn7CF2P4EJ198YJ8jy2kI4X3ZPTmaeQk3PKag1MxDkbcFPHa/5Nvr5ycnujMP5JdVMvPsAKkEOxz7Prrnj+cMwOZxdZV1ldEGW166L+mHvulFIhHe0ThJieW/91cswkOXPua4a1FB47lxGJ8QEkEOPWG9+GAU4Y/wh7QEtO9mjDrJ0aPLB7+hjoEfOM0BTQE6eMHYxTo88KnkKryu8c6pacrF9OnjQLQGZelihC3J/ojLE/WVVLWRiGi7ZNef7bqs1SzGkqPHaN3lP1oFMPazQBbmzvIubLLoUBbnjDMKYfLADFng9apqGOVLXYVkH8qTCirT6dVL9VQdkOWlZY9SyQgsJR9Nb2Qri47ZOKDROPaP3aBXIkworEs44xEw1+XlBzvtwjivZG1cJLuEshFLPcDzBCjEMeKu1O9sHudV6nWVjhCzIJoQVeUGWP6uqYkwS5DyJ5iQs6RBQ14LvxZ1OUkwCZOrVirwgdzYnPL8MHcoI2di8c03VSVgTWphQrcgDsu+KjejyKU1uXQ/vvJUNsiq272/SEHYS1oDci5uoQywhkZD2ttBVsdyeu+J2t6iVZ0jSgNIFOdyOlmdXwrMVjWov9gw3HIhraw3InmDnVHeeIklTOykhlFWC5IQn/RB8kS1qp1b5+mRsvPZDdQyAJMhhyTA9Njai/AbDeeQiOgFkgAyQIThABsgwdVUkciYDumbTCSADZIAMwQEySZCLHKiB4ACZBMhFD9RAcICM0AIGkAEyQIZOiJEBMmJkgAyQEVrAADJABsjQCSADZIAMkAEyQIYBZJtBtuGdGGX1dy4TZFt7I08A5NWlOTb3d9X3lzL/jqH9fKOgxXs9pzcsVNlH8fKU8fe/iuiatY8zBdMdq0qnZwPZ1H6+UVPW3zUbBxYFOWsfZzLhgeZYyYBsaj9fpTeurJ2V1SiwDI/ch4Q4yEXGSsojj/LSJoHc6eITTAKv/ZSvVygalwJkC0Cm3s93xEPXftLyKYjxZf8ygDyFIMf6+RLv/xZPWBpvPGflU6/zZa8ZH0A2FmS99qcm9PPNeS/aL/Mp67+obQeZdFvZSfXzTVoZdI+wPkn+wlKcXuXFVJAn3XOafDuAcffzTRKmyBHW+N/SqyObCHLeHslTAfKgEmBWLTn+MAZJ3xR5ZIQWozwa4X6+mSaGO1c64wfIBoJsaj/f6D3EV5TBi14AsuUgm9zPdxjYilg7k2dD5PjlQyg/W/GaW0VWozJeT5bWx5mSI9AZKxmQTe7n209yBlWWcPwSarfpLxQNq4q9VD29jzOtXEJvrGhiSNzQjdOiZA8TBJABMkAGyAAZIANkCA6QrQO5+xpWG/4Xb5ym0igR5CnVNatOpYIsGLvTfTXXtHmUrOcRplnXpHMcYwMZBntWzw0RYAAZBiNi/wGtMgDoqS2hVwAAAABJRU5ErkJggg==)
=![](data:image/png;base64,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)
=
=![](data:image/png;base64,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)
R.H.S = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAI0AAABACAYAAAAnKPTPAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAPESURBVHja7Z2xattAGMevc5MtS0rOWztksy8ZA5lUkSbdCrVFtuKhuIKSQUMpF2+G4hfI1Gx9hILzAvUTtJCMnZJnqOuzLFmWJVmyTyfp9A/8IYQg3333u/u+01l/kX6/TyAoixAECNBAiqGZ/jyLE4JVAxhSjr3/C+eX+4yQh+k//VsVfTRt3kRg9VXPoFfRY08m1OhdxULTIq3RJef7CCIUhAnQQMVBo7LOQX2lNt7SobHtdtMw2FdKyZ/wxfLQl4/vXrUoGYvcOhNtjY1270396g32UEQW2Bqaa4udk0bjtwAmqkCSLW6bTUrIU7gwE2LW9XkdgFnEoKLQLMGTMzSDQXfHPCB31LR4dzDYcQPYfmlSMnLBKSaISoERY+KvsjWFJqoBSTOMGfaHOJgIoU/h2wDBvK9D/dNhdCgmjdvfaGjy7nOloEkOJrkNBlGAZJnHn0XaFGnLtswzRsmvWRtZZ+itVFWSiLFo+2KSLEOjqs9aQOMHkXVulyFya50Ge31jWvbZooCsXv0jxuKIHv0QYxEHjcXIdxV91gIatzBsjcPtjAqWV0Sq2OXJTkte6o2DRlWflUMTvr/iNWCb+y4ioFGzKCmAwVWpqHssaRQ1SJtCI6vPSqGJOcuYzLX097RLadJKVVZoEs50VtQ4ObkR9UgQJMfp7vrQOM6L4CTTDhrZ6Ul8ZtRuqgorjQTAApON3XtjBGjWbb+ZxVMEvNLQJBb+dUhPsqDxtp7h/C+OFxgzv3lbS0BTYmjEgPE2u5gtl9OGZSlks0JjW8fv58cIkyh51xJt8Lbdwev77Tww77qOs6sTNHF99ia0rD5vDc2c4sdwzk0LQhZoBDDJhaP7xbBZUN2jhaX2rNQH1BxV8Sbfon+LWkZln/F9GgjQQIAGAjSQDpoV1qGdGKCBAA0EaCBAAwEaCNAAGgjQQIAGAjQQoIEADQQBGgjQQIBGB+no2bMGmtPDPbL395TzQwCQTTp79nxiz4eEvv0JaGTPxLJ49rT5BaCpGjwaevaUEhpdPGVUQFOEZ0+poNHNU0YFNEnKy7OnVNDo5ilTJDRRnj2y/GvWQKN+y62Tp0yR0OTp2VO6pxF0ev56U88eGe+iyNOzB9AUDE0WrxoZjzoDGqSnyM/M27MH0GgEjSrPHkCjCTQqPXtKBY1OnjKiL74vTEbPnqza1rMnq39NaaDRyVNmW8+erMCo9uyBAQC0UQkBaCBAAwEaCNBA9YBmbqWO90bWT1HjvhYaRsg93std3xUl7TkYAgZlX5EQBAjQQLnrP+WnpQr/FY8ZAAAAAElFTkSuQmCC)
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
∴ L.H.S = R.H.S
And so on for other values of n.
If we notice each result then we will see that it is of same type that we are trying to prove.
So we can generalise the above results for all positive integer values of n.
![](data:image/png;base64,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)
Hence Proved
Question 4.If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
Answer:To prove: AB – BA is a skew symmetric matrix.
Symmetric matrix: A symmetric matrix is a square matrix that is equal to its transpose. In simple words, matrix A is symmetric if
A = A’
where A’ is the transpose of matrix A.
Skew Symmetric matrix: A skewsymmetric matrix is a square matrix that is equal to minus of its transpose. In simple words, matrix A is skew symmetric if
A = -A’
Given: A and B are symmetric matrices i.e.
A = A’ …(1)
B = B’ …(2)
Now calculating the transpose of AB – BA,
(AB – BA)’ = (AB)’ – (BA)’
(By property of transpose i.e. (A – B)’ = A’ – B’)
= B’A’ – A’B’
(By property of transpose i.e. (AB)’ = B’A’)
= BA – AB
= -(AB – BA)
Or we can say that: (AB – BA) = - (AB – BA)’
Clearly it satisfies the condition of skew symmetric matrix.
Hence AB–BA is a skew symmetric matrix.
Question 5.Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Answer:Case 1: When A is a symmetric matrix i.e.
A = A’……. (1)
where A’ is the transpose of A
To prove: B’AB is also a symmetric matrix.
Calculating the transpose of B’AB
(B’AB)’= B’A’(B’)’ (By property of transpose i.e. (AB)’ = B’A’)
= B’A’B (By property of transpose i.e. (A’)’ = A)
= B’AB (from (1))
It satisfies the condition of symmetric matrix as matrix B’AB is equal to its transpose.
HenceB’ABis a symmetric matrix when A is symmetric.
Case 2: When A is a skew symmetric matrix i.e.
A = -A’……. (2)
where A’ is the transpose of A.
To prove: B’AB is also a skew symmetric matrix.
Calculating the transpose of B’AB
(B’AB)’= B’A’(B’)’ (By property of transpose i.e. (AB)’ = B’A’)
= B’A’B (By property of transpose i.e. (A’)’ = A)
= B’(-A)B (from (2))
= - (B’AB)
It satisfies the condition of skew symmetric matrix as matrix (B’AB) is equal to its transpose.
Hence(B’AB)is a skew symmetric matrix when A is skew symmetric.
∴ Both results are proved.
Question 6.Find the values of x, y, z if the matrix satisfy
the equation A’A =I.
Answer:Given A ![](data:image/png;base64,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)
Transpose of a matrix: If A be an m×n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. It is denoted by A’ or AT.
∴ Transpose of A = A’ =![](data:image/png;base64,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)
Given equation
A’A =I
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
=![](data:image/png;base64,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)
. =![](data:image/png;base64,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)
As these matrices are equal to each other that means each element of matrix on L.H.S is equal each element of matrix on R.H.S.
∴ On comparing elements on both sides we get
4y2 + z2 = 1 …… (1)
2y2 – z2 = 0 …… (2)
x2 + y2 + z2 = 1 …… (3)
– y2 – z2 = 0 …… (4)
From equation (4) we get,
x2 = y2 + z2 …… (5)
Substituting this value in equation (3) we get,
2y2 + 2z2 = 1 …… (6)
Subtracting equation (2) and (6) we get,
3z2 = 1
z2 = 1/3
z =
1/3
Substituting value of z in equation (2) we get,
2y2 = 1/3
y2 = 1/6
y = �1/6
Substituting values of y and z in equation (5) we get,
![](data:image/png;base64,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)
x2 = 1/2
x = 1/2
Hence values of x, y, z are 1/2, 1/6, 1/3 respectively.
Question 7.For what values of ![](data:image/png;base64,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)
Answer:Multiplying matrices on the left hand side
L.H.S = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= [6.0 + 2.2 + 4.x]
= [4 + 4x]
R.H.S = O (O
is the null matrix)
= 0
∴ L.H.S = R.H.S, so we get,
[4+4x] = O
4 + 4x = 0
4x = -4
x = -1
Hence value of x is equal to -1.
Question 8.If
show that A2 – 5A + 7I =0.
Answer:To prove: A2 – 5A + 7I = 0
Given: A = ![](data:image/png;base64,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)
L.H.S: A2 – 5A + 7I
R.H.S = 0
I = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAABACAYAAACqaXHeAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAHpSURBVHja7duxbsIwEAbg29uRBQmPXdiI1Z0pihDqXBqxMlEvzJVhy9KH6FtUKk/AGzAwduMdqEIaVIHBEF+gTv7h1pD7Yp+dnKHZbEZ1jlonDwAA/AHQetiURCsi2hyGWEdKd3xJahyKiTkP2ohwPDkKEFDwNdS6WdWnncIAAABMAEoNOmEo34Sg5f5Fyw4VR71A0GI7r0WwOLc+sQFMY9knIZZp8qbCUmZsfzstyrHq5RiCaC3jaf/qUyC7mesB5CvV/u9lVV+ubPfuPUCW6OGSrFXUSUeB7T68BkiS0X3Uojm1ovkoSe5NI4Pky0dlAXZJGgBO4VQH4HeYm57yDsBSBwBQiSlgADg1PVAEq7cMHg7zfHrYNkP+b4SOrPc33wjZhl6ZmyEVPz5ffSu8q8iWjw3lvQxlSecvQwOlH/A6DAAAAOCsor1XsAEAAAAAAAAAAAAAAAAAAAAAADUH6LYb1Pjuat0+52JFe/QcUfRswqu8eyfx9OkM4NKjZ3mKBc8msAC49uhZIW4B4Nqj9xqAoz3lNQBHj/4fA9iXQY4W9S0BnL8KA4ChR+81gO9FkKUx4tqj9x7AtUfvPYBpM3RJj54b4JIpx3tYumCP3jVcziagOwwAAAAAAAA4AVDr/w3WNQBQd4AfoliatvvW+KUAAAAASUVORK5CYII=)
Calculating value of A2:
A2 = A.A
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
Substituting value in L.H.S we get,
= A2 – 5A + 7I
.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0 = R.H.S
L.H.S = R.H.S
Hence A2 – 5A + 7I = 0 is proved.
Question 9.Find x, if ![](data:image/png;base64,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)
Answer:Multiplying matrices on the left hand side
L.H.S = ![](data:image/png;base64,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)
= 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)
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= ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFYAAAAlCAYAAADY4B6YAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAOISURBVGje7Zq/T9tQEMfPHZiAiaGUXDZUxIZf2BlQeFGhW9omVrogeYiQB4jkAbUOWwbozFbSpVH3LvyYOrX9E4CxC+RfaOg72w+cnzXFMU76kG5IFPDzx3ff+94R2NvbAxXRh4KgwI4wWPHzpF8oWC4fbQAjrSdYxynNMoBL8YHf3YFX3HKWHnKoL2PwkCyO2735QAuzWzt9weqgn5QcZzbqA9m2OcURTugQGrCLvG3PjVMmb2Vx51HAWnx5U2a8weAIUvzUrNUmFdgIw7H4EoL+Y9jX+f/AulrOjhXYIWTsMrc2H7Orh+z42siApcManG8+lr4KABUAdtl5j6aZny4yPECAK9npkRUP8qY5nXiwBJUOwErVF/J1nFB3y6u6ANfsBEuOZS0Fp4D8pGA5815VFeZdF5NaOzVteyqxYMnDUraIi1eozGw7P5fNWq/igmrbpWcM4buo9VYn2KrB1oVPb3b6dK/JQpMZ1fUHgl1ZnIGZXyuOs/iXzOs5cQya0ooMPkY9cNynUtwy58Z7TpnZAdYz/dhcLe/q3U1Wu5QVNihKCxP1iYVS/Z/B+tPZuffk4UacuYXc2vbfv/Deiw9aqGZZYBvIjP1azZzsBdbPzGvQ33ySmkoPg36P5CFMP3gwWAo6IJl8F24qeyY1yD0IsHOpU0kIkoAMZj4TyH5gAxXVAtR/5sv2c7qXdDrzNey9RAJWws0hHFOGsoKzQV01h9hIUqZKCZBnGgTWmwhxH/xKRL34IUzTihxsQBYu6KCZDDbCaFFEW6QnYfym21CETAWToR9Y+huua0D8RpaLqpHghrVbkYK966jtkjDkLVJXBOG1WSsBJvgAqKqyKTgjsLQEkk3Wt4EVxFyD9JSsFlWjdo/7ihys2xjEU9a8tVkl6eu9u6ZLr9k5Za7XG4SkBWxVsI902qihg/VKJ9ewqtZTbyXoWhaW1Hm+nxSIxlXv5WNvP8+K9djAksGX3bbNsvhlNkpg/ey+WSq8exnU69uJjBlHsYClC5JW+VOUFtQpt8yELpFlGRWw0puLrL3Ovi6vSz02ODpSLmIBG5yiZOPo1rRkDQkS7JprEbthubuBTPoweA/pDD+M3ceqaI+3CxNHA8DGs48dx0jEoluBVaHAKrAKrIKlwI4KWLn5UV+KC7/OLHv/z+sP1hvthvOluHGIQetM5OXtnmBVRJzdCoICO1LxB1BA2lmukcgUAAAAAElFTkSuQmCC)
R.H.S = O
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAgCAYAAADqgqNBAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFvSURBVFjD7ZdJbsIwFIZfWXMBEF72AKnFDVBkROgWuVa2LKLIEoOUBaqc7HKM9hzlAOUO5QKEK3Swm4imaZIOsikCIr1FrOT//Pz7eYAoiuC/Ag4KLp8LGY18/En4BzpffgoZduSHrzKe00AbwoX1W3jo4sGHBrxInW1RpxKOWejoGl7fRrPjgRc93Rtcgb2R7WAEq3dPEV71vAVW7cbhguKhFEsIm/TV+4SRPgJIMBVDo/AgcNsyxTUifJpv5wRNAfDaDYK2MbgUmpcJCU4smf0W2f7MCDyOx03SgSV0yHIcx81PcOG21IgAvrk3At8BSuB1HdMDz4a2LLsdXPkuROvI4DW+1lmib8IheKifcOzOWKlV1fPC610pS8r+0wbPfE9kPc/zy21a//ip6Lf25TXNHiXZet7waZfKDm2wGw72sqtx1h0pYLaxPFIuLk9jPz/Dz3CtcIveXuu6NFQdQL477Ou6NJTqHNZd7WTgb+9QcDPjdUpuAAAAAElFTkSuQmCC)
∴ L.H.S = R.H.S we get,
![](data:image/png;base64,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)
x2 – 48 =0
x2 = 48
![](data:image/png;base64,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)
Hence the required value of![](data:image/png;base64,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)
Question 10.A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWcAAABfCAAAAAARjVklAAAAAXNSR0IArs4c6QAABkJJREFUeNrtndF56yoQhKcFaqAF9UAJqoEW1AEdqAMqUAU0oAbogB7mPAAysqxzLTnJkXJ3n8jnZI1+42WRZxxQ4icCguBinAPcRS8B9+EcAJtHFghHOMN83+QCAKBz6STnCy2NB2dEkoy4HGegS7+Hs84L2qnuSpwNyUn9Hde9OBuLSCY1GAQyGgUoG/Nk5x4IedKpgycZekC7ZcW57+TMCYaESVbBkVMHdBNJMg0K2gLLbxawoQe0jS5PLjANGjBjugTnCEc6RIOw8FOJDOiAhbOFJ+nzw/ZnOAcYEloBcPWph/ya53jiXH7FLJy7OrgCZ/YqUVuaZT7RYMxbZKzXkDFH1U3k3GH+ibpBD0MCJpCMUJ70CjM5oJvJ6ZlzBMbE6Pq6vD36SE72IpwDnEdsOOd5BgzLDwM8SY51z3Q/wDl5BU+iI0m6im4glUq1Precx+XtVTiP5QquUZ9Jo7Rl4RydMbkg1M0kwPky7Mv79fs51/q0PI/JL3GC4YyerzgbzGvOEejHcB3OAQiFs3+AfHDukJczTeVsfoKz8Ww5L21OndiW83O/MWkAyl2FM7Vh5pyAIcRaNypnjErN7cX+TF+3ep5mPYdytNpyjpu+Lk6DKovkApzLlYRak584D5wzaIfxX3Fu6nOCfvTPDWf3XJ9zTP+0mX7NeYaO5GSe6jM5oUtkBFwkwzCS6hvbpRec237DwCbOPcA8jeTUut/Il8HRBjK5C3IuFVhtONOjS6vybb+/f169b5r+eS6TBMncK+tV/0xqAAjucRi4Gudkgc6FLWcOsCRDr8oJK1kF7X+Oc3seDB2Ujfnc7RS0L3OcekAPkZx7oJuj6wBt4z3ui0oIZ+EsIZyFs3CWEM7CWUI4C2fhLCGchbNwlhDOwllCOAtn4SwhnIWzxAHOpnwQf72YFn3I1AHd+pN1p4E+bMbJqiLerhGsBvTagNHm20m0RFqkE1OPp9yrRHvjq3OeFnuB30qtbaNrbsZFF93qNaoBo/njNt9OoiXmvj7vWLQg8+tEe+NH3bika8k9rjgpjOQIPJZSgApMFvpp7NBFxq46m0hSjzPpgemxQpt8e4mW51lwpax16otU9TnR3vhGnH2+NNtI+8pYY16PNeasEHtONzTvhjbfXqIt56rcad77baK98dU5k3SF85DnOzWrtMssBvjVOBXAHeIm19gwf+TbSdTW98I55ooRm1m0ifbGN+JcbAYrKRjKb7jVOCwS3meFZUu+zbeT6AVneqD3U9fU/jbR3vj/xXnlnzjJOZcynSic9zj7VUN1kvOAYbIKehbOO5w9+vQx54CeZHJNO/LLOH+6D9q2zTu9D1b83aMd+WX74Gd9XTLPWvgDfd2Kcy7xzcO/rK9btf1+c6T4+zklavVYnnFzjPiPc0rD2QMuMfb5YQ1/8Jxy0XO3W4zFz+dkd+jcXY2IwGM/fP/cXf86PB4tXbRKB8/dN+Dc3pYpTti37yOtOPeF4dv3kVrO9GbxYFTj7YH7SPeK+WlPOxZKf9E0erxvebkl5xHxkxfpq1xL6sCrfUvOpv+kFqkvmkVYtyS/kPMNQzgLZ+EsIZyFs3CWEM7CWUI4C2fhLCGchbOEcBbOwllCOAtnibc4V13Bd34L/KmIVmP96faeBGArB2g9KXP/gT/lpCflRpw9noUle5KWrbyl9aTMapPofZ3MWU/Ky7pRxCjhUmqZKX8xb7tIdyRaL+RarSfFwCaG9iug39d9nfak3Iez3kjy9ySHr+WH1ZNSNI2+QfW+jvG0J+U2nKet7mhPQvtaTls9KVO+/tSs0vd1uac9Kbfh7OCsgupjW3TrI+/Iw6sK2jVf5c+6I72tMz/rSbkN5/r/Fpod6Cjn7En5jPNZT8qNONuZnLumrB7kXDS4n3E+60m5EeeQCyROcq6elI84n/ak3IazLT7hBs+hfXDxpHy0D572pNyGc2mT5gbPgb6u8aR81Ned9qTchnNScImhg1tsJe+fU1aelPacctSfctqTcp9zt2+OzmVLe/vcvfJKtOfuw/6Us56UG93fqP9PsrGVvHsfacW5vY902J9y1pNyz/uiX2Yr+ShRf9KycRvOX2Yr+SyROulAug3nL7OVfJTokCflnnXj3iGchbNwljgafwCPntC0OzTVxAAAAABJRU5ErkJggg==)
(a) If unit sale prices of x, y and z are Rs. 2.50, Rs. 1.50 and Rs. 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs. 2.00, Rs. 1.00 and 50 paise respectively. Find the gross profit.
Answer:(a) Given the unit sale prices of x, y and z as Rs. 2.50, Rs. 1.50 and Rs. 1.00 respectively.
Unit sale prices can be represented in form of matrix as: ![](data:image/png;base64,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)
Calculating total revenue in market I:
Number of products in the form of matrix:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAM4AAAAgCAMAAABtj7cEAAAAAXNSR0ICQMB9xQAAASNQTFRFgYGBAAAAAAAAiIhmZoiId3dmZnd3ZmZVVWZmf39dXX9/d3ddXXd3bn9Zf25ZWW5/d2pVVWp3f25ISG5/e2pEampVRGp7RGp7cVtISFtxW25ISFtugF0zbFlGRllsM12AgFszRldqM1uAf1kzM1l/bls1bFk1NUhubkgdHUhubEYdHUZsbEYdHUZsRkYzHUhbMzNbW0gdMzNIWzMdWjMdHTNaHTJaSDQeHjRIWzMAWTMAADNbWjMAADNaADJaADJYHR1JHR00SR0ASB0AAB1ISBwAABxIABxIHR0dAB0zMx0AAB0zHQAzMh0AAB0yAB4eHh4AHR0AMwAAAAAzMwAAAAAzMwAAAAAzHQAAAAAdHQAAAAAdHQAAAAAdAAAAAAAAAAAAOK+07AAAAGB0Uk5TAAECDAwNDQ4OGRkaGiUlJScnMTEyMjIzSUlKSktLS0tMTExNTVdZY3Jyc3N0dH5+fn+LjI2NjpmZoKCgoaGioqi2uru7vLy9w8jJycnKytfY2dvb3Nzd3erq6+vs7P3+Gum/CwAAAnZJREFUeNrtmGlzmlAUhs8FYgJtNCXUdLENBZO4pWtqm1qbtgbTNd1SpWou/P9f0ZH1AvcyYqYzxeF88fnA+855FBAFKOY/H8RxHLc6Aqpt4T/bOdapYWxdboc6Wu7PsNbq6yAOEWdjnBhImfmpjJaOR7dAi9UkddDNhz/a3nH3PuHPVRQhBlJm66ttn9+nZnbimJQJt1g7MjF+XmJmyZqkTu3bd9sr0qe34dakHiEGJkeZYIwtO5m5S8HkZR1swRunFSgPhiVWlqyhnWyqV3Rt1ASAxvg6QQxMDj94JEB5YI2lWKZNQdr4W6jOhvJMY2QjNWk6Ha+oTRADkyMfuG/t/Jgl4uEWjcsqAIgjzdvVCfh4CNCaBpiqwxtDwSnqhwR0ZM/rsQTLxX0deXKyAUg/FSjZiwim64gj50DeGG4GJIhUZNrwRo8syhL3deCVdX5DfVdZqCZFR5713Y3GdwKSZCoydZQvElmUJR7owDPL7pYWq/m3OuhoF66sg5QPptXduKqOd1aLo+FmQIJIRZZNqxktyhL3dVDnrVB+b52VGDUXZE3arWDgXXq9kICOjFEPIFqUJe7r6PO1+GOrvUhN2o3avZcrM40gBlI/G+Wx80LJyDFM0zl2NuSNPiMbqUnTkSeH88/6t0QQA2k2enf+0Lb1VFgqHr6pdh0BrBk9dg0Kaig6nG6/cR/5GtMqt2/uRoiBSZv9me1MM57ZoSDl+dXfQvw5fcCtP/kl+QEtzMaRoiObGGPsXMOwZ+KPlRgxMD572B33B2HmOLFF+QXG+GVloZpW8fOt0Cl0Cp3V1qnn/m/DTqhTC74l8vu3Yb4Fisnh/AXDVQYJbFi09QAAAABJRU5ErkJggg==)
So the total revenue is given by:
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMwAAAAgCAMAAABpemc5AAAAAXNSR0ICQMB9xQAAAVZQTFRFgYGBAAAAAAAAiIhmZoiId3dmZnd3ZmZVVWZmf39dXX9/d3ddXXd3VXd3ampqbn9Zf25ZWW5/antVe2pVd2pVVWp3f25ISG5/e2pEampVRGp7ampVVWpqRGp7cVtISFtxW25ISFtugF0zbFlGRllsM12AgFszRldqM1uAf1kzM1l/bls1NVtubFk1NUhuW0g1bkgdHUhubEYdHUZsbEYdHUZsSDNIRkYzHUhbMzNbW0gdNDRIMzNIWzMdHTNbWjMdHTNaHTJaSDQeHjRIHjNHMx5HWzMAWTMAADNbWjMAADNaWjIAADJYHR1JADNIHR00HR0zSR0ASB0AAB1ISBwAABxIABxIHR0dMx0AAB0zMx0AAB0zHQAzMh0AAB0yHQAyAB4eHh4AHR0AMwAAAAAzMwAAAAAzMwAAAAAzHQAAAAAdHQAAAAAdHQAAAAAdAAAAAAAAAAAAD12a2wAAAHF0Uk5TAAECDAwNDQ4OGRkaGhskJSUlJiYnJzExMjIyMzMzSUlKSktLS0tMTExNTVdXWWNwcnJzc3R0fH5+fn+Ki4yMjY2OmZmbm6CgoKGhoqKorra4uru7vLy9w8jIycnJysrK19jZ29vc3N3d6urr6+zs/f58ypyoAAADT0lEQVR42u2Y61sSQRTGz+wuJGsllK5kmWZKXNK4djcyy5AsE6GLJV0JNsDZ+f+/9MzsbZadRepT+DAf4OUw7+H8ZmbPAgCT8d8OJEmSdFYAEsTAv+fHGGYFY+Nk3oZJjv3pyp9xGHr2EKcl50jCgOKlYEinWIa6kV0DcssRermgH2b2mJDmqmmfbhNCenQGunmEPy0gj+KlYFz+gHFzzpm3iIKlH+XKve8FpkJlHeOn4UAvn8UHE+9ijA2SNt/FGOMKVaneElzvpj2Kl/4Rf6RA9Kijsnk3XItI+q/mr98Ig5HrhzGI1hrhIC+fZRBGrt1XIFozWBHTL8LWwk23cwCQ7ZznFC/tBXXXWb6rAIDWT9J5Bdcikn43rYfBJFh9wWk8WQZhtA1zPVj4FXm7ap7HopWzwCle2oe2MLg93TUIsJzitmCyJwsAEGkn7VK1fgkg37Olq4K72S7dGU3HBmleYHANheWsuooLBsJIuxXeDH/htmC07t4UoNSh4hhaVU5ywUAYuc4uFLh4RzfoDkbazCPXGzOOUtygIoZBV4/fhHmzEhndbcHAS6M5l9iPBaS55Mkihol/Vm2sGqnSbayajJ1lR6luUBWWIx9gg2yFObOqje62YWCTJQFxmmVPFiEMKq85OtLqqKPAsK9GxRJ/kwI5o5PSSDB+tw2D4u91Y2vqn2FQPsctUfZk3j7ZkXZjxlGKG6RbnKVd3DDoo7sSoHU7akRoCXQnPTCo+FqJHhjvwsB7W045LT6LCCaxwZ987YsKcs26Wiuu4oLB3Yx2ErHlNLcFk6JlyTtGQez1ZvHDoPgD9mRnpZ3EauTxfpJTvAyEKR4q1jzNtQzKITA7rD65Xg3werL4YFBqi57c2ceKeRNDZdbIuyW65b9UTvFSUA77xhSq5yDAMtzt3GdIGtE0lUEvcspBTpZBGLTeJ2zkAMrP56TQ5m3zyuktSOv6mkfx0ldOtNW8JoU2n5jzFl2LSAqWIkX2aC+I/Ojdks49/Kk6hiR4JBf0wWSwOejvtIyO8XbMyp7R8cfYgOKliex2DvkZxnh/CYZYhrg1nRZBX0dpnu0hn8wF85MfZxOYCcwE5qzBpMf+T8CiDbNi3VzG+U/A8QaYjLEafwC9a0dY7g3GBAAAAABJRU5ErkJggg==)
=![](data:image/png;base64,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)
∴ Total revenue in market is Rs 46000.
Calculating total revenue in market II:
Number of products in the form of matrix: ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMYAAAAgCAMAAAB+WPfwAAAAAXNSR0ICQMB9xQAAASNQTFRFgYGBAAAAAAAAiIhmZoiId3dmZnd3ZmZVVWZmf39dXX9/d3ddXXd3bn9Zf25ZWW5/antVd2pVVWp3f25ISG5/e2pERGp7RGp7cVtISFtxW25ISFtugF0zbFlGRllsM12AgFszRldqM1uAf1kzM1l/bls1bFk1bkg1NUhuNUhbNUhbbkgdWUgzHUhubEYdHUZsbEYdHUZsHUhbMzNbW0gdMzNIWzMdWjMdHTNaHTJaSDQeHjRIWzMAWTMAADNbWjMAADNaADJYHR1JHR00SR0ASB0AAB1ISBwAABxIABxIHR0dAB0zMx0AAB0zHQAzMh0AAB0yHh4AMwAAAAAzMwAAAAAzMwAAAAAzHQAAAAAdHQAAAAAdHQAAAAAdAAAAAAAAAAAA5JdEJQAAAGB0Uk5TAAECDAwNDQ4OGRkaGiUlJSYnJzExMjIzSUlKSktLS0tMTExNTVdZY2NwcXJycnNzdHR+fn+LjI2NjpmZoKCgoaGiqLa6u7u8vL3DyMnJycrK2Nvb3Nzd3erq6+vs7P3+pkT0lwAAArZJREFUeNrtmFt30kAUhWeSlJYgIA3Qi9giUKwiN2+ViogXCNqqqDXQAJP8/1/hSgLJhDmTZtWXxsV+4VuzZh9mMycTEoQ2umPCgiAI0V950TTIdTaKMQqEGIusG6MU3VZq/NcxsCDgVdt5DYgYhGR1Kg7whK2E3TqwhTaDMfD2s5+kZ8HDS/Itj31EI6TMD9McH0GevQBktXU2JeRNjGvxmaEYuKpfHEgWlWcHaF+v+IhGQDmdEGKYjOdBMDIS1WEapQajGM/iMwMxcMuoO5TQLKhN7lFEI/Ttg+cSSg2MiczxJLQmgKyK9sKUeYlj8ZuBGOVVCtRalmpSRCMg5dT5KRdZjidspdoijxCKa6XVGu15HjZmDrU5MRLaMOZcPKI6kuxSfY+owaCT4/1Ehj2hKyn6hx2Ey0MJtqzoqs+J0TKfnhNjvGvNtqeK6ijpkuQNSvwUotqj3NKNCNV4Z4x3i5/SCC50329mYojq4omE9vVFHinzvjMyOXRJ9gZlfozcd5lyyzciWOS1YXZiduMAlkO/mYmxDFwz+7ePgc+O0T/HwLmvU6Ozc8sYy5lxbSIvuzaujZIuSd4gt6lwo+71/JonHrYSbn2UUp+NLzG4UPLKZwZ2w40pDpYXVM8jRCFPRee0Aj2hK5Wt5Yhdowlb1szstTGwt0lUh9LyVM7NSxTRCO9F7oX9wXoUPrLq2gsT1T7H4jezJ1XBqGDrZmy1t962dvePTBGNYIpyx/pTlXklwZ6wlWqmtYwttcexKHobe2bg9tedHQmZS3uzarO8UJ0e+4hGIEV1btqqr3n2gpFR/NfskbD98rfszishH1KD8H+qx1NyfergyZRcpNeIRkYnxJH9AAZ7QlZKnRNC3gZ8OW1ubB6bNjE2MTYxIhWjEt3XbS03RmF12kfxdVtEV77Rnddf3/IEPZ1gXG4AAAAASUVORK5CYII=)
So the total revenue is given by:
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=
.
∴ Total revenue in market is Rs. 53000.
(b) Given the unit cost prices of x, y and z as Rs. 2.00, Rs. 1.00 and 50 paisa respectively.
Calculating gross profit in market I:
Unit cost prices can be represented in form of matrix as: ![](data:image/png;base64,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)
So the total cost of products in market I is given by:
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
Since the total revenue in market I is Rs. 46000, the gross profit in this market is given by:
(Rs. 46000 – Rs. 31000)
= Rs. 15000.
Calculating gross profit in market II:
The total cost of products in market II is given by:
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
=![](data:image/png;base64,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)
Since the total revenue in market II is Rs. 53000, the gross profit in this market is given by:
(Rs. 53000 – Rs. 36000)
= Rs. 17000.
Question 11.Find the matrix X so that ![](data:image/png;base64,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)
Answer:Given X![](data:image/png;base64,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)
From above equation it can be observed that matrix on R.H.S is a 2×3 matrix and that on the L.H.S is also a 2×3 matrix. Therefore, X must be a 2×2 matrix.
Let X ![](data:image/png;base64,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)
So the equation is given by:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now equating the corresponding elements of both the matrices we get,
a + 4b = -7, 2a + 5b = -8, 3a + 6b = -9
c + 4d = 2, 2c + 5d = 4, 3c + 6d = 6
Now, a + 4b = -7
⇒ a = -4b -7
∴ 2a + 5b = -8 ⇒ 2.(-4b -7) + 5b = -8
⇒ -8b -14 + 5b = -8
⇒ -3b = 6
⇒ b = -2
∴ a = -4b -7 ⇒ a = -4.(-2) -7
⇒ a = 1
Now, c + 4d = 2 ⇒ c = -4d + 2
∴ 2c + 5d = 4 ⇒ 2.(-4d + 2) + 5d = 4
⇒ -8d + 4 + 5d = 4
⇒ -3d = 0
⇒ d = 0
∴ c = -4d + 2 ⇒ c = -4.0 + 2
⇒ c = 2
Thus, a = 1, b = -2, c = 2, d = 0.
Hence X becomes![](data:image/png;base64,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)
Question 12.If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn = BnA. Further, prove that (AB)n = AnBn for all nϵN.
Answer:To prove: ABn = BnA
Given A and B are square matrices of same order such that AB = BA.
We have to prove it using mathematical induction.
Steps involved in mathematical induction are-
1. Prove the equation for n=1
2. Assume the equation to be true for n=k, where kϵN
3. Finally prove the equation for n=k+1
Let P(n): ABn = BnA
For n=1,
L.H.S: ABn = AB1 = AB
R.H.S: BnA = B1A = BA = AB
So, L.H.S = R.H.S
∴ P(n) is true for n=1.
Now assuming P(n) to be true for n=k, where k ϵ N
P(k): ABk = BkA …… (1)
Now proving for n=k+1, i.e. P(k+1) is also true
L.H.S = ABn
= ABk+1
= (ABk).B
= (BkA).B …… from (1)
= Bk(A.B)
= Bk(BA) (∵AB = BA)
= Bk+1A
R.H.S = BnA
= Bk+1A
∴ L.H.S = R.H.S
All conditions are proved. Hence P(k+1) is true.
∴ By mathematical induction we have proved that ABn = BnA.
Now, to prove: (AB)n = AnBn for all n ϵ N
For n=1,
L.H.S = (AB)n = (AB)1 = AB
R.H.S = AnBn = A1B1 = AB
∴ L.H.S = R.H.S
∴ It is true for n=1
Assuming it to be true for n=k then,
(AB)k = AkBk ……(2)
Now proving for n=k+1,
L.H.S = (AB)n
= (AB)k+1
= (AB)k(AB)1
= (AkBk)AB
= Ak(Bk.A)B
= Ak(A.Bk)B (ABn = BnA)
= (AkA)(BkB)
= Ak+1Bk+1
R.H.S = AnBn
= Ak+1Bk+1
∴ L.H.S = R.H.S
All conditions are proved. Hence P(k+1) is true.
∴ By mathematical induction we have proved that (AB)n = AnBn for all nϵN
Hence proved.
Question 13.
Answer:Given A = ![](data:image/png;base64,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)
Calculating A2:
A2 = A.A
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKQAAABACAYAAACUYNzVAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAWqSURBVHja7Z09b9tGHIdP8djEU5akOm1pgWzWWXuGgKaRl82IKUJTUA2CyiEJwMGIaW8GmmZOurTp1PYLtIiTvf0EbRF7axcrH0F1yiNFiZKOer3THamfgT9gSDJEPXqO92b+SI6PjwkKZUoBAgpColAzCRn+XMmqtQVESGkCl5Lg9WC4BMP+L0HQuMEIOQ9f8N940QvbC7bWEaZn0ydiJuSSWu2n6deCobhaMcPuLAyHYFZJ9bQRBDfQdUyvtkWfioQEw+UYAiaEhJAaxy4lCGk2w8LCbDb3NutV+pIScpGM4Wi1/nKv2dyEkLMz3B8w7MYM90OG/iaEnKNOTppX7TJ5Rwg7c7zgVvT5POdWOOE4I2XrfdP3r0HIhRl+UMmwkDADz94KW/XH0Q975LL7YUv/xNyj+xDSTIYLwxwdWwgeL+mCmQUtcNiD8LAuWePonilCmspxKkOThBwdn1W27dfJ2ExmC5IFM/myHUbeELr7tnlyctUEIUfGaEZxTL9/utGoZjg3TN9vXrPK5H04wO0kC72eW3tE6e5PfFwRvsmz8LmPOheB0zB7Xc8FqVT+JJT9sef7n5swqZmRY0cXR10M54YZgyKfqO09GfoADXavbNePIshl+52MFiS7u4l3Xdj5XuvgC91Cms5xOkP/S+1CDmZe42fA6DlKTvmHIMz9QeeAPHP8k2ztSfiilxEyDxynMDwLGS49015ayMFerbhL5m+ge/w4CeYkEVYpZB446mK4oJDsXPS6epX8qHv8OG1CUOeDcmOEnMixY8o4fJUM5++yo+5k/GB8v3FTVndY9DNkHjjOwLCjXchkUFsi5HLLef4wvWYWD9JLl3xlv+H7N9PPaYPpBA/S79//sql9qnMMOcoxOUaTOCYM08e3CoYLte5dSt7yFnK3dcD42lTbqTmUkA5fn3JtesjfhD9+0LpbrdneY11Ckkrlr5rr7fO1Pb7m58RdzYUJC+MJx9KA40YGxw0dHHUxXBimy+iL9D+fUua+4C0mOqVvV15Fj1P2u45ttHR349m1x/1/sAiPJ9mX1S1kwtGJOXYncOzq4JjBsKua4cIwTa5V7HIUfS9bF0PAhJAQUnXF63hql5+KLqQuhoWEWWfke9UXVUVnEFZ/U1QhdTEsJMyVdWkFFlIXQ8CEkBASQqIgJISEkBASBSEhJIQETDCEkBASQqIgJISEkBASsCAkhISQYAghISSELEhFGTcQUjrDFMw7t6+T6//eCYLbgDW9vmaffUvow9+GhQTDZRkCJoSEkBASJV3IQUTbL1cEzwkfN6Fk5THKEDKvDGVxlCpk22Jf8YTV0XjffkyI5sApUcnMtZQhZB4ZJhzT2eOLcpQmZD8Bq3ddNrXaz4ZmTppzaUQlO9eyB/NXgZD/FJVhwnFnJEpl0VxLEcOFliwimMw95Be084NJcmj6uS8GtmzZeYzLXnWYR4ayOSq56pAD3Kmw7/jf8cgPSujfqlv2lHvnjd1LT0Ueo8zLYHUwNIGjsstgeawKP8DoWl4JIUTTasL9B8eKt2QVeYyyr8teNUNeE+5BKODYks5RmZB8IMsPgucGygghkl0qci1lC2k6w+FxbwbHOIhq5rGvMiF5KDqzrMPtiv1adzZkVpcoO49RtpCmM1TBUZ2QvZZjastOunmZuZbShcwBw1GOy+ZaKhOStxyLWc9NbdnJMcrMtZQtZB4Y9idgwlzLUshxZ65cS2VCRjNDav+cB5iyci1lC5kXhgnHOqPfDOdaOnPnWioTMjpdG5AtvuL1OKlCgqHEdchobQowwdAEIZO7hpq6s5AHIcFQppDx7PBDzfUeQUgw1C7kulbRE3SNEZLf6VO0l7mu0LL2e1vhJCRLyB7DDTCcn+FYtyHey1Qb7WtyTdo35/u7oq5XvD+8vgwn7ZmPMkTXgTLrbAoIKAiJQmXU/z+NxVTwW+o6AAAAAElFTkSuQmCC)
=![](data:image/png;base64,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)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMEAAABCCAYAAADuZrLqAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAd3SURBVHja7V09b9tGGD7ZY2NnyZJYpy0okCnm2XuGgKaRj42NJUGTUSIQXAKxHHAIbNmbgHzMTZfEndr8gRZ2vLf9AS2aymOH2PkJTVQdKVofJiUe+R7Jk97hGSzJ4nMP+ZD33r16X3JwcEAQiFkGioBAE6AICDTBwB8mIfMkHAUUDKEKOt3rNfxa7hQCTdBs1q4zQk67H/rvMkofjcb+bRQXoQq2DPok6FruOuMz1bcaoSbQiHZcazavp0X0HSFznjPNeTxxiFTModNGbkzgONbCOiVHnju1tuk4S3iSEDNlgi1jddOfYlUZeUuKxnur1bqCJwoxMyYYRNM2linR/sji2Ag0QT5M4Abl7CiNY/fjEB/v5mbtQlBZg6Tcc2uCZ/W72qphb0qPQyxzcUOjryghZ95qAT2j2sYr03IWZ8UAKmsAwT2XJuDruVXD2JQdD7Ra1hWjSE4IYe2y3bzpTcPKNxkh/5CifmI5zsK0G6CnwXsVNYDinjsTcANwUqy2f88lI7gh522IdCL9jxd3kE+jAuxX2f3ucTusun9/2k2gsgZQ3KWaYHiu1p+nkf7OXWH08491+pTqj5/y9x3HXNJ1+5uk87swhInVLLMHXWpffCMGGy147mlmMJ8W1RlCgzzwn8g9axOMztVKK8Zr07IWx5GvMPIm6a50XBMMCD5fZuSQ0PWjsOnYls6+5Xcg75HbunjkOk7thsZ32Fn1bVZzYk9nZzHqHXG8BmtHsqekSfhDcZdigpZjLejuXJue+xexXV19ROn6T3ye1j3ojvue3VxOY7krygWw3zBulwj5SEqlPwuU/R62UeetWhXaPaPy7fad4TuQnHFN1Ll3TFGd42gAZoCE/Mdyt+1ipoGxR550qGFvD5GusXtFY2PfHbikjTCI6ZBt0O0C0U7NuvN1oAlYeY9z5yes0BvHRZCW4lMAQuc4GuSFfxj3uiB3cBP0I3b6adTB7nuUHHPiEBdLUFbgYEwxKeM1dE7pJw4WjZNxFxAfz1qJ/cD14Uu6lNC/U3sKAOkcSYOAVZbxGZnjMzSh+E/g3g7jLt0E/czTy4MbdD/EqgN3fFBGYPf8fB59PSjACxNx3AkaRZXRF/wz7riocZxWmkcEnRtRdI6rQZD2YaBGfVsG/7HcXRNNPn+STcBOg/6vopEfZc6bIaZDvSD9MIqI/DtYufmgwuYO01hJgdYZQoME/NtJ+ENxlzMdCnHhxeqJxMQ4CBMM3AUnXkR8rZrp+t5KyXidZrIflM4QGmTFH4q7lMDYC0zIl+Xy7kN/Lk68TbAdvn5bIKxdc5wb/bXgaJtbskwwyNMdt2MuuSehN73pz38v8/TvaGk+BZLoPDqWqBrkkf8w905s7nKWSHnASMlRoevEu/VnjJPeKrMKJYVzvn5bNegePyh/vRtQMsgcoTgmIKXSX6tVe8O0rKuWZV4te4/SM//C9uenQd/Lx6ozfS+LlO8hnS1nJYrOo2OJqkEW/MsT+ENxl7ZZxgdYYfT54MYXZZXn/nKivlL63n2dst8g0zL43SUoEJv0OOW/ZXDXmXuc/FyUwSAw6HvdVSFq/JzV7x5EdR4dS1QNsuevXeIfzl0T4p7bLNI0AJEf4z66Ff7xj8p5UlDc+Z7I4KYnmkB8rf4ETaA2d/d7WOVwJk3Q25aPvQToZzGmuUssYzVN1jKoKtxn2gRewl780jGePuRDNyh7pLwGCpoAivtMmwCBQBMgEGgCBAJNgECgCRAINAECTYAmQKAJ0AQINAGaAKEeIEv4owkQyoFXqVgbKuHfXEITIGYKIyX83yRNYEQTIJQGRAl/NAFCbRMAlPBHEyACgk11ehRAlPB3KweGm+DOrWvk2r93ms1beJFMebCpYI8CqBL+37GvXhL68Fc0wQxDxR4FSUv4owkQAcElTI+CDnAJnbBpW12nO4Ml/A3DNjM1QdJ6/Unq60PcUVTlDjUWyB4FIiVv4uoYWMI/wa/LuiZ40TXBL4kC4yT1+uP0MYAEPHcnNe5QY4HsURDHBFnrmLjkSqR6/SG/4c2yj4Hq3GONJU6df8EeBaImcAB6LOTDBDHr9WfZx0B17sFjqe7GGQtkjwJRE+RBR9DiWyL1+qHq6yetk58ld5mrPSJ9EyB7FIj0h8iLjuAV6KLW64fqY5C0Tr4k7o2sC1qJ9E2A7FEg0h8iLzqCmyBqvf6s+xhMG/e4Y5kUfIr2KBCZDkH0KMilCaLW68+6j8G0cY87lohPgnMZJsiLjrI61QTW6x+tjZ9lHwMR7kH9CfLGPcl5gOxRIBoY50FHKZ1qwur1j9bGz7KPgQj3oP4EeeMeMpbdKOcBskeBqAni9CjIvQnG1esPqvOfVR8DEe5h/QnyxD3JeYDsUSDSH0JcR02KjuAmULlev+q9BuKOReXy7Lkzgcr1+qeh18ClgBZNkL4JVK7XPw29BuKOReUeBfkzgcL1+qeh10DcsajcoyCXMQECMZUmMG27GJKPU0AREaogLM/M+4HOWBOQD8G5OPHbHCEQmdzxDfokal4ZCoaYeaAICDQBioCYdfwPp/U5QjEoiLkAAAAASUVORK5CYII=)
=![](data:image/png;base64,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)
And given that A2 = I
Then ![](data:image/png;base64,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)
Comparing corresponding elements we get,
![](data:image/png;base64,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)
Question 14.If the matrix A is both symmetric and skew symmetric, then
A. A is a diagonal matrix
B. A is a zero matrix
C. A is a square matrix
D. None of these
Answer:Given A is both symmetric and skew symmetric matrix then, A = A’ and also A = -A’
⇒ A’ = -A’
⇒ 2A’ = 0
⇒ A’ = O
Clearly it is observed that transpose of A is a null matrix or zero matrix then matrix A must also be a zero matrix.
Hence A is a zero matrix.
Question 15.If A is square matrix such that A2 = A, then (I + A)3 – 7 A is equal to
A. A
B. I – A
C. I
D. 3A
Answer:Given that A2 = A
Calculating value of (I + A)3 – 7 A:
⇒ I3 + A3 + 3I2A + 3IA2 – 7A
⇒ I + A2.A + 3A + 3A2 – 7A (In = I and I.A = A)
⇒ I + A.A + 3A + 3A – 7A (A2 = A)
⇒ I + A2 + 3A + 3A – 7A
⇒ I + A - A
⇒ I
Hence(I + A)3 – 7 A = I.
Let, show that (aI + bA)n = anI + nan-1bA, where I is the identity matrix of order 2 and nϵN.
Answer:
To prove: (aI + bA)n = anI + nan-1bA
Proof: Given A
We will be proving the above equation using mathematical induction.
Steps involved in mathematical inductionare-
1. Prove the equation for n=1
2. Assume the equation to be true for n=k, where kϵN
3. Finally prove the equation for n=k+1
I is the identity matrix of order 2,
i.e. I
Let P(n): (aI + bA)n = anI + nan-1bA, n ϵ N
For n=1,
L.H.S: (aI + bA)1 = aI + bA
R.H.S: a1I + 1a1-1bA= aI + a0bA= aI + bA
So, L.H.S = R.H.S
∴ P(n) is true for n=1
Now assuming P(n) to be true for n=k, where k ϵ N
P(k) : (aI + bA)k = akI + kak-1bA …… (1)
Now proving for n=k+1, i.e. P(k+1) is also true
L.H.S = (aI + bA)k+1
= (aI + bA)k . (aI + bA)1
= (akI + kak-1bA). (aI + bA) ……from (1)
= aI(akI + kak-1bA) + bA(akI + kak-1bA)
= aI(akI) + aI(kak-1bA) + bA(akI) + bA(kak-1bA)
= (a.ak) (I×I) + kb(a.ak-1)(IA) + (bak)(AI) + (bb) kak-1(AA)
= ak+1 I2 + ka1+k-1 bA + bakA + b2kak-1A2 (IA = AI= A & I2 = I)
= ak+1 I + kak bA + bakA + b2kak-1A2
Calculating A2
A2 = A.A
∴ A2 = O (O is the null matrix)
Putting value of A2 in L.H.S
L.H.S = ak+1 I + kak bA + bakA + b2kak-1(O)
= ak+1 I + kak bA + bakA + 0
= ak+1 I + kak bA + bakA
= ak+1 I + (k+1)ak bA
Putting n=k+1 in R.H.S
R.H.S = ak+1 I + (k+1)ak bA
∴ L.H.S = R.H.S
All conditions are proved. Hence P(k+1) is true.
∴ By mathematical induction we have proved that P(n) is true for all n ϵ N.
Thus,(aI + bA)n = anI + nan-1bA, where I is the identity matrix of order 2 and nϵN.
Question 2.
If , prove that
.
Answer:
We will be proving the above equation by putting different values of n (i.e. n = 1, 2, 3 ….n)
For n=1,
For n=2,
A2 = A.A
For n=3,
A3 = A2.A
r n=4,
A4 = A3.A
d so on for other values of n.
If we notice each result, then we will see that it is of same type that we are trying to prove.
So we can generalise the above results for all n ϵ N
Hence Proved
Question 3.
If then prove that
where n is any positive integer.
Answer:
We will be proving the above equation by putting different values of n. Because n is a positive integer so it will take values which are greater than 0 i.e. n = 1, 2, 3 …n
For n=1,
L.H.S = An = A1 = A =
R.H.S =
∴ L.H.S = R.H.S
For n=2,
L.H.S = A2
= A.A
∴ L.H.S = R.H.S
For n = 3,
L.H.S = A3
= A2.A
∴ L.H.S = R.H.S
For n = 4,
L.H.S = A4
= A3.A
=
==
R.H.S =
=
=
∴ L.H.S = R.H.S
And so on for other values of n.
If we notice each result then we will see that it is of same type that we are trying to prove.
So we can generalise the above results for all positive integer values of n.
Hence Proved
Question 4.
If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
Answer:
To prove: AB – BA is a skew symmetric matrix.
Symmetric matrix: A symmetric matrix is a square matrix that is equal to its transpose. In simple words, matrix A is symmetric if
A = A’
where A’ is the transpose of matrix A.
Skew Symmetric matrix: A skewsymmetric matrix is a square matrix that is equal to minus of its transpose. In simple words, matrix A is skew symmetric if
A = -A’
Given: A and B are symmetric matrices i.e.
A = A’ …(1)
B = B’ …(2)
Now calculating the transpose of AB – BA,
(AB – BA)’ = (AB)’ – (BA)’
(By property of transpose i.e. (A – B)’ = A’ – B’)
= B’A’ – A’B’
(By property of transpose i.e. (AB)’ = B’A’)
= BA – AB
= -(AB – BA)
Or we can say that: (AB – BA) = - (AB – BA)’
Clearly it satisfies the condition of skew symmetric matrix.
Hence AB–BA is a skew symmetric matrix.
Question 5.
Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Answer:
Case 1: When A is a symmetric matrix i.e.
A = A’……. (1)
where A’ is the transpose of A
To prove: B’AB is also a symmetric matrix.
Calculating the transpose of B’AB
(B’AB)’= B’A’(B’)’ (By property of transpose i.e. (AB)’ = B’A’)
= B’A’B (By property of transpose i.e. (A’)’ = A)
= B’AB (from (1))
It satisfies the condition of symmetric matrix as matrix B’AB is equal to its transpose.
HenceB’ABis a symmetric matrix when A is symmetric.
Case 2: When A is a skew symmetric matrix i.e.
A = -A’……. (2)
where A’ is the transpose of A.
To prove: B’AB is also a skew symmetric matrix.
Calculating the transpose of B’AB
(B’AB)’= B’A’(B’)’ (By property of transpose i.e. (AB)’ = B’A’)
= B’A’B (By property of transpose i.e. (A’)’ = A)
= B’(-A)B (from (2))
= - (B’AB)
It satisfies the condition of skew symmetric matrix as matrix (B’AB) is equal to its transpose.
Hence(B’AB)is a skew symmetric matrix when A is skew symmetric.
∴ Both results are proved.
Question 6.
Find the values of x, y, z if the matrix satisfy the equation A’A =I.
Answer:
Given A
Transpose of a matrix: If A be an m×n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. It is denoted by A’ or AT.
∴ Transpose of A = A’ =
Given equation
A’A =I
=
=
. =
As these matrices are equal to each other that means each element of matrix on L.H.S is equal each element of matrix on R.H.S.
∴ On comparing elements on both sides we get
4y2 + z2 = 1 …… (1)
2y2 – z2 = 0 …… (2)
x2 + y2 + z2 = 1 …… (3)
– y2 – z2 = 0 …… (4)
From equation (4) we get,
x2 = y2 + z2 …… (5)
Substituting this value in equation (3) we get,
2y2 + 2z2 = 1 …… (6)
Subtracting equation (2) and (6) we get,
3z2 = 1
z2 = 1/3
z =1/3
Substituting value of z in equation (2) we get,
2y2 = 1/3
y2 = 1/6
y = �1/6
Substituting values of y and z in equation (5) we get,
x2 = 1/2
x = 1/2
Hence values of x, y, z are 1/2, 1/6, 1/3 respectively.
Question 7.
For what values of
Answer:
Multiplying matrices on the left hand side
L.H.S =
= [6.0 + 2.2 + 4.x]
= [4 + 4x]
R.H.S = O (Ois the null matrix)
= 0
∴ L.H.S = R.H.S, so we get,
[4+4x] = O
4 + 4x = 0
4x = -4
x = -1
Hence value of x is equal to -1.
Question 8.
If show that A2 – 5A + 7I =0.
Answer:
To prove: A2 – 5A + 7I = 0
Given: A =
L.H.S: A2 – 5A + 7I
R.H.S = 0
I =
Calculating value of A2:
A2 = A.A
=
=
=
=
Substituting value in L.H.S we get,
= A2 – 5A + 7I
.
= 0 = R.H.S
L.H.S = R.H.S
Hence A2 – 5A + 7I = 0 is proved.
Question 9.
Find x, if
Answer:
Multiplying matrices on the left hand side
L.H.S =
=
=
=
=
=
=
R.H.S = O
=
∴ L.H.S = R.H.S we get,
x2 – 48 =0
x2 = 48
Hence the required value of
Question 10.
A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below:
(a) If unit sale prices of x, y and z are Rs. 2.50, Rs. 1.50 and Rs. 1.00, respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs. 2.00, Rs. 1.00 and 50 paise respectively. Find the gross profit.
Answer:
(a) Given the unit sale prices of x, y and z as Rs. 2.50, Rs. 1.50 and Rs. 1.00 respectively.
Unit sale prices can be represented in form of matrix as:
Calculating total revenue in market I:
Number of products in the form of matrix:
So the total revenue is given by:
=
=
=
=
∴ Total revenue in market is Rs 46000.
Calculating total revenue in market II:
Number of products in the form of matrix:
So the total revenue is given by:
=
=
=
=.
∴ Total revenue in market is Rs. 53000.
(b) Given the unit cost prices of x, y and z as Rs. 2.00, Rs. 1.00 and 50 paisa respectively.
Calculating gross profit in market I:
Unit cost prices can be represented in form of matrix as:
So the total cost of products in market I is given by:
=
=
=
Since the total revenue in market I is Rs. 46000, the gross profit in this market is given by:
(Rs. 46000 – Rs. 31000)
= Rs. 15000.
Calculating gross profit in market II:
The total cost of products in market II is given by:
=
=
=
=
Since the total revenue in market II is Rs. 53000, the gross profit in this market is given by:
(Rs. 53000 – Rs. 36000)
= Rs. 17000.
Question 11.
Find the matrix X so that
Answer:
Given X
From above equation it can be observed that matrix on R.H.S is a 2×3 matrix and that on the L.H.S is also a 2×3 matrix. Therefore, X must be a 2×2 matrix.
Let X
So the equation is given by:
Now equating the corresponding elements of both the matrices we get,
a + 4b = -7, 2a + 5b = -8, 3a + 6b = -9
c + 4d = 2, 2c + 5d = 4, 3c + 6d = 6
Now, a + 4b = -7 ⇒ a = -4b -7
∴ 2a + 5b = -8 ⇒ 2.(-4b -7) + 5b = -8
⇒ -8b -14 + 5b = -8
⇒ -3b = 6
⇒ b = -2
∴ a = -4b -7 ⇒ a = -4.(-2) -7
⇒ a = 1
Now, c + 4d = 2 ⇒ c = -4d + 2
∴ 2c + 5d = 4 ⇒ 2.(-4d + 2) + 5d = 4
⇒ -8d + 4 + 5d = 4
⇒ -3d = 0
⇒ d = 0
∴ c = -4d + 2 ⇒ c = -4.0 + 2
⇒ c = 2
Thus, a = 1, b = -2, c = 2, d = 0.
Hence X becomes
Question 12.
If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn = BnA. Further, prove that (AB)n = AnBn for all nϵN.
Answer:
To prove: ABn = BnA
Given A and B are square matrices of same order such that AB = BA.
We have to prove it using mathematical induction.
Steps involved in mathematical induction are-
1. Prove the equation for n=1
2. Assume the equation to be true for n=k, where kϵN
3. Finally prove the equation for n=k+1
Let P(n): ABn = BnA
For n=1,
L.H.S: ABn = AB1 = AB
R.H.S: BnA = B1A = BA = AB
So, L.H.S = R.H.S
∴ P(n) is true for n=1.
Now assuming P(n) to be true for n=k, where k ϵ N
P(k): ABk = BkA …… (1)
Now proving for n=k+1, i.e. P(k+1) is also true
L.H.S = ABn
= ABk+1
= (ABk).B
= (BkA).B …… from (1)
= Bk(A.B)
= Bk(BA) (∵AB = BA)
= Bk+1A
R.H.S = BnA
= Bk+1A
∴ L.H.S = R.H.S
All conditions are proved. Hence P(k+1) is true.
∴ By mathematical induction we have proved that ABn = BnA.
Now, to prove: (AB)n = AnBn for all n ϵ N
For n=1,
L.H.S = (AB)n = (AB)1 = AB
R.H.S = AnBn = A1B1 = AB
∴ L.H.S = R.H.S
∴ It is true for n=1
Assuming it to be true for n=k then,
(AB)k = AkBk ……(2)
Now proving for n=k+1,
L.H.S = (AB)n
= (AB)k+1
= (AB)k(AB)1
= (AkBk)AB
= Ak(Bk.A)B
= Ak(A.Bk)B (ABn = BnA)
= (AkA)(BkB)
= Ak+1Bk+1
R.H.S = AnBn
= Ak+1Bk+1
∴ L.H.S = R.H.S
All conditions are proved. Hence P(k+1) is true.
∴ By mathematical induction we have proved that (AB)n = AnBn for all nϵN
Hence proved.
Question 13.
Answer:
Given A =
Calculating A2:
A2 = A.A
=
=
=
=
And given that A2 = I
Then
Comparing corresponding elements we get,
Question 14.
If the matrix A is both symmetric and skew symmetric, then
A. A is a diagonal matrix
B. A is a zero matrix
C. A is a square matrix
D. None of these
Answer:
Given A is both symmetric and skew symmetric matrix then, A = A’ and also A = -A’
⇒ A’ = -A’
⇒ 2A’ = 0
⇒ A’ = O
Clearly it is observed that transpose of A is a null matrix or zero matrix then matrix A must also be a zero matrix.
Hence A is a zero matrix.
Question 15.
If A is square matrix such that A2 = A, then (I + A)3 – 7 A is equal to
A. A
B. I – A
C. I
D. 3A
Answer:
Given that A2 = A
Calculating value of (I + A)3 – 7 A:
⇒ I3 + A3 + 3I2A + 3IA2 – 7A
⇒ I + A2.A + 3A + 3A2 – 7A (In = I and I.A = A)
⇒ I + A.A + 3A + 3A – 7A (A2 = A)
⇒ I + A2 + 3A + 3A – 7A
⇒ I + A - A
⇒ I
Hence(I + A)3 – 7 A = I.