Question

If $A=(\begin{array}{cc}3& 2\\ 5& 1\end{array}),B=(\begin{array}{cc}1& -2\\ 2& 3\end{array})$ and $O=(\begin{array}{cc}0& 0\\ 0& 0\end{array})$ then
verify (i) A + B = B + A
(ii) A + (-A) = O = (-A) + A.

Answer 1

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

$\Rightarrow \left(i\right)A+B=B+A$

$L.H.S=A+B=\left[\begin{array}{cc}3& 2\\ 5& 1\end{array}\right]+\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]$

$=\left[\begin{array}{cc}4& 0\\ 7& 4\end{array}\right]$

$R.H.S=B+A=\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right]+\left[\begin{array}{cc}3& 2\\ 5& 1\end{array}\right]$

$=\left[\begin{array}{cc}4& 0\\ 7& 4\end{array}\right]$

$\therefore L.H.S=R.H.S$

$\Rightarrow \left(ii\right)A+(-A)=0=(-A)+A$

$A+(-A)=\left[\begin{array}{cc}3& 2\\ 5& 1\end{array}\right]+\left[\begin{array}{cc}-3& -2\\ -5& -1\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$A+(-A)=0$

$(-A)+A=\left[\begin{array}{cc}-3& -2\\ -5& -1\end{array}\right]+\left[\begin{array}{cc}3& 2\\ 5& 1\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$(-A)+A=0$

$\therefore A+(-A)=0=(-A)+A$

Hence, verified.