Question

If $A=(\begin{array}{cc}8& -7\\ -2& 4\\ 0& 3\end{array})$ and $B=(\begin{array}{ccc}9& -3& 2\\ 6& -1& -5\end{array})$, then find AB and BA if they exists.

Answer 1

The matrix A is of order $3\times 2$ and B is of order $2\times 3$. Thus, both the products AB and BA are defined.
Now, $AB=(\begin{array}{cc}8& -7\\ -2& 4\\ 0& 3\end{array})(\begin{array}{ccc}9& -3& 2\\ 6& -1& -5\end{array})$
$=(\begin{array}{ccc}72-42& -24+7& 16+35\\ -18+24& 6-4& -4-20\\ 0+18& 0-3& 0-15\end{array})=(\begin{array}{ccc}30& -17& 51\\ 6& 2& -24\\ 18& -3& -15\end{array})$
Similarly,
$BA=(\begin{array}{ccc}9& -3& 2\\ 6& -1& -5\end{array})(\begin{array}{cc}8& -7\\ -2& 4\\ 0& 3\end{array})=(\begin{array}{cc}78& -69\\ 50& -61\end{array})$ (Note that $AB\ne BA$)