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?

Let, $B = \begin{bmatrix} a & b\\ c & d\end{bmatrix}$ 
$ A*B= \begin{bmatrix} 0 & 0\\ 0 & 0\end{bmatrix}$ results in a system of linear equations (in this case 4 equations and 4 variables). 
We know (0,0,0,0) is always a solution but it is also possible that the above set of equations are over determined (true in your example) in which case you have multiple solutions. Hence A and B don't have to be zero matrices. Also note that if you have linearly dependent rows or columns, the resultant system of equations is over determined.
