Question

Show that product of two given matrices can be a zero matrix without either of the matrices begins a zero matrix.

Answer 1

Let us choose two non-zero matrices A and B as under
$A={\left[\begin{array}{cc}1& 1\\ 3& 3\end{array}\right]}_{2\times 2}andB={\left[\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right]}_{2\times 2}$
$AB=\left[\begin{array}{cc}1(-1)+1.1& 1.1+1(-1)\\ 3(-1)+3.1& 3\left(1\right)+3(-1)\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& -0\end{array}\right]=O$
Thus we observe that the matrix AB is null matrix whereas neither A nor B is a null matrix.
$BA=\left[\begin{array}{cc}2& 2\\ -2& -2\end{array}\right]\ne O$