Verify A(BC) = (AB)C, if A = [101230045],B=[2-2-1103]

Exercise 2.3 | Q 4 | Page 55

QUESTION

Verify A(BC) = (AB)C, if A = ${\displaystyle \left[\begin{array}{ccc}1& 0& 1\\ 2& 3& 0\\ 0& 4& 5\end{array}\right],\text{B}=\left[\begin{array}{cc}2& -2\\ -1& 1\\ 0& 3\end{array}\right]}$

SOLUTION

BC = ${\displaystyle \left[\begin{array}{cc}2& -2\\ -1& 1\\ 0& 3\end{array}\right]\left[\begin{array}{ccc}3& 2& -1\\ 2& 0& -2\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}6-4& 4-0& -2+4\\ -3+2& -2+0& 1-2\\ 0+6& 0+0& 0-6\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}2& 4& 2\\ 1& 2& -1\\ 6& 0& -6\end{array}\right]}$

∴ A(BC) = ${\displaystyle \left[\begin{array}{ccc}1& 0& 1\\ 2& 3& 0\\ 0& 4& 5\end{array}\right]\left[\begin{array}{ccc}2& 4& 2\\ -1& -2& -1\\ 6& 0& -6\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}2-0+6& 4-0+0& 2-0-6\\ 4-3+0& 8-6+0& 4-3-0\\ 0-4+30& 0-8+0& 0-4-30\end{array}\right]}$

∴ A(BC) = ${\displaystyle \left[\begin{array}{ccc}8& 4& -4\\ 1& 2& 1\\ 26& -8& -34\end{array}\right]}$     ...(i)

AB = ${\displaystyle \left[\begin{array}{ccc}1& 0& 1\\ 2& 3& 0\\ 0& 4& 5\end{array}\right]\left[\begin{array}{cc}2& -2\\ -1& 1\\ 0& 3\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}2+0+0& -2+0+3\\ 4-3+0& -4+3+0\\ 0-4+0& 0+4+15\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}2& 1\\ 1& -1\\ -4& 19\end{array}\right]}$

∴ (AB)C = ${\displaystyle \left[\begin{array}{cc}2& 1\\ 1& -1\\ -4& 19\end{array}\right]\left[\begin{array}{ccc}3& 2& -1\\ 2& 0& -2\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}6+2& 4+0& -2-2\\ 3-2& 2-0& -1+2\\ -12+38& -8+0& 4-38\end{array}\right]}$

∴ (AB)C = ${\displaystyle \left[\begin{array}{ccc}8& 4& -4\\ 1& 2& 1\\ 26& -8& 34\end{array}\right]}$       ...(ii)

From (i) and (ii), we get
A(BC) = (AB)C.