Question

If $A=(\begin{array}{cc}3& 2\\ 4& 0\end{array})$ and $B=(\begin{array}{cc}3& 0\\ 3& 2\end{array})$ then find AB and BA. Are they equal?

Answer 1

$A=\left[\begin{array}{cc}3& 2\\ 4& 0\end{array}\right],B=\left[\begin{array}{cc}3& 0\\ 3& 2\end{array}\right]$

$\Rightarrow AB=\left[\begin{array}{cc}3& 2\\ 4& 0\end{array}\right]\left[\begin{array}{cc}3& 0\\ 3& 2\end{array}\right]$

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

$=\left[\begin{array}{cc}15& 4\\ 12& 0\end{array}\right]$

$\Rightarrow BA=\left[\begin{array}{cc}3& 0\\ 3& 2\end{array}\right]\left[\begin{array}{cc}3& 0\\ 3& 2\end{array}\right]$

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

$=\left[\begin{array}{cc}9& 6\\ 17& 6\end{array}\right]$

$AB\ne BA$

Hence, the answer is $AB\ne BA.$