Find a, b, c, if [135ab-5-7-4c0] is a symmetric matrix.

Exercise 2.2 | Q 7 | Page 47

QUESTION

Find a, b, c, if ${\displaystyle \left[\begin{array}{ccc}1& \frac{3}{5}& \text{a}\\ \text{b}& -5& -7\\ -4& \text{c}& 0\end{array}\right]}$ is a symmetric matrix.

SOLUTION

Let A = ${\displaystyle \left[\begin{array}{ccc}1& \frac{3}{5}& \text{a}\\ \text{b}& -5& -7\\ -4& \text{c}& 0\end{array}\right]}$

∴ A^T = ${\displaystyle \left[\begin{array}{ccc}1& \text{b}& 4\\ \frac{3}{5}& -5& \text{c}\\ \text{a}& -7& 0\end{array}\right]}$

Since A is a symmetric matrix,
A = A^T

∴ ${\displaystyle \left[\begin{array}{ccc}1& \frac{3}{5}& \text{a}\\ \text{b}& -5& -7\\ -4& \text{c}& 0\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}1& \text{b}& -4\\ \frac{3}{5}& -5& \text{c}\\ \text{a}& -7& 0\end{array}\right]}$

∴ By equality of matrices, we get

a = – 4, b = ${\displaystyle \frac{3}{5}}$, c = – 7.

Concept: Algebra of Matrices

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board

Chapter 2 Matrices

Exercise 2.2 | Q 7 | Page 47