Matrix Algebraic Operations, If AA = AB, does A = B? A and B are 2 x 2 matrices and A is not a zero matrix.  How is the following proof incorrect?
Since AA = AB, AA - AB = 0
A (A - B) = 0 and since A does not equal zero, then A - B = 0, therefore A = B.
Thank you

$$\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}=\underbrace{\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}}_{=A}\underbrace{\begin{pmatrix}3 & 5 \\ 0 & 0 \end{pmatrix}}_{=B}$$
EDIT Here's an example where the product is not $0$.
$$\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}=\underbrace{\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}}_{=A}\underbrace{\begin{pmatrix}2 & 3 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}}_{=B}$$
