If A = [2-35-4-61],B=[-122203]and C=[43-14-21], Show that A + B = B + A

Exercise 2.2 | Q 1.1 | Page 46

QUESTION

If A = ${\displaystyle \left[\begin{array}{cc}2& -3\\ 5& -4\\ -6& 1\end{array}\right],\text{B}=\left[\begin{array}{cc}-1& 2\\ 2& 2\\ 0& 3\end{array}\right]\text{and C}=\left[\begin{array}{cc}4& 3\\ -1& 4\\ -2& 1\end{array}\right]}$, Show that A + B = B + A

SOLUTION

A + B = ${\displaystyle \left[\begin{array}{cc}2& -3\\ 5& -4\\ -6& 1\end{array}\right]+\left[\begin{array}{cc}-1& 2\\ 2& 2\\ 0& 3\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}2-1& -3+2\\ 5+2& -4+2\\ -6+0& 1+3\end{array}\right]}$

∴ A + B = ${\displaystyle \left[\begin{array}{cc}1& -1\\ 7& -2\\ -6& 4\end{array}\right]}$       ....(i)

B + A = ${\displaystyle \left[\begin{array}{cc}-1& 2\\ 2& 2\\ 0& 3\end{array}\right]+\left[\begin{array}{cc}2& -3\\ 5& -4\\ -6& 1\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{cc}-1+2& 2-3\\ 2+5& 2-4\\ 0-6& 3+1\end{array}\right]}$

∴ B + A = ${\displaystyle \left[\begin{array}{cc}1& -1\\ 7& -2\\ -6& 4\end{array}\right]}$       ....(ii)

From (i) and (ii), we get
A + B = B + A.