Question

If $A=\left(\begin{array}{cc}3& 2\\ 4& 0\end{array}\right)$ and $B=\left(\begin{array}{cc}3& 0\\ 3& 2\end{array}\right)$ then, find $ABandBA$.

Answer 1

$AB=\left(\begin{array}{cc}3& 2\\ 4& 0\end{array}\right)\left(\begin{array}{cc}3& 0\\ 3& 2\end{array}\right)$

$=\left(\begin{array}{cc}3\times 3+2\times 3& 3\times 0+2\times 2\\ 4\times 3+0\times 3& 4\times 0+0\times 2\end{array}\right)$

$=\left(\begin{array}{cc}9+6& 0+4\\ 12+0& 0+0\end{array}\right)$

$\therefore AB=\left(\begin{array}{cc}15& 4\\ 12& 0\end{array}\right)$

Again $BA=\left(\begin{array}{cc}3& 0\\ 3& 2\end{array}\right)\left(\begin{array}{cc}3& 2\\ 4& 0\end{array}\right)$

$=\left(\begin{array}{cc}3\times 3+0\times 4& 3\times 2+0\times 0\\ 3\times 3+2\times 4& 3\times 2+2\times 0\end{array}\right)$

$=\left(\begin{array}{cc}9+0& 6+0\\ 9+8& 6+0\end{array}\right)$

$\therefore BA=\left(\begin{array}{cc}9& 6\\ 17& 6\end{array}\right)$