If the product of two matrices, $A$ and $B$ is zero matrix, prove that matrices $A$ and $B$ don't have to be zero matrices I can give an example where product of two non-zero matrices is zero matrix,
$$
A=
        \begin{bmatrix}  
        3 & 6 \\
        2 & 4 \\
        \end{bmatrix}
$$
$$
B=
        \begin{bmatrix}
        2 & -8 \\
        -1 & 4 \\
        \end{bmatrix}
$$
the product is zero matrix.
I don't know how to prove this. Could someone help?

Like
$$ \begin{vmatrix} x^2 & x \\ x & 1 \end{vmatrix} \begin{vmatrix} -x & -1 \\ x^2 & x \end{vmatrix} = \begin{vmatrix} 0 & 0 \\ 0 & 0 \end{vmatrix} $$
for any finite $x$.
