Can a regular square matrix times any matrix of the same size equal zero?
If you multiply a matrix by its inverse you get the identity matrix...
$\displaystyle \left(\begin{array}{cc}1 & 2\\ 3&4\end{array} \right).\left(\begin{array}{cc}-2 & 1\\ \frac32&-\frac12\end{array} \right) = \left(\begin{array}{cc}1 & 0\\ 0&1\end{array} \right)$