Show by an example that for $A \ne 0,B \ne 0$ and $AB = 0$.

Let $A = \left[ {\begin{array}{cccccccccccccccccccc}0&{ - 4}\\0&2\end{array}} \right] \ne 0$ and $B = \left[ {\begin{array}{llllllllllllllllllll}3&5\\0&0\end{array}} \right] \ne 0$

$\therefore$ $AB = \left[ {\begin{array}{llllllllllllllllllll}0&0\\0&0\end{array}} \right] = 0$

Hence proved.