If $A$ and $B$ are $2\times 2$ matrices and $AB=0$ then $A=0$ or $B=0$? 
Is it true that if $A$ and $B$ are $2\times 2$ matrices and $AB=0$ then $A=0$ or $B=0$. Prove it, or prove the contrary.  

I tried saying that if:
$$A= \begin{pmatrix}
        0 & 0  \\
        0 & 0  \\
        \end{pmatrix}\quad\text{and}\quad B= \begin{pmatrix}
        e & f  \\
        g & h  \\
        \end{pmatrix}.$$
The product will be 0. But, if $A$ is non zero, i tried doing something like this:
$$A= \begin{pmatrix}
        a & b  \\
        c & d  \\
        \end{pmatrix},\quad
    B = \begin{pmatrix}
        e & f  \\
        g & h  \\
        \end{pmatrix}
\implies
AB = \begin{pmatrix}
        ae+bg & af+bh  \\
        ce+dg & cf+dh  \\
        \end{pmatrix}$$
is equal to \begin{pmatrix}
        0 & 0  \\
        0 & 0  \\
        \end{pmatrix}
But then, how should i proceed? I don't know if these are the right steps to follow, i've tried searching, but i didn't find the right examples.

For example you may try with
$$A=B=\begin{bmatrix}
        0 & 1  \\
        0 & 0  \\
        \end{bmatrix}.$$
