if $A\times B = A$, is $B ={} $Identity matrix?

Question

Let $A, B ∈ M_{n×n}$

If $A\cdot B = A $, then $B$ is the identity matrix.

I can't find a theorem that proves this statement. Could it be false?

Edit:

A is NOT the zero matrix.

Answer 1

A counterexample: $$A=\begin{bmatrix}1&1\\0&0\end{bmatrix},\quad B=\begin{bmatrix}0&1\\1&0\end{bmatrix}.$$ $AB=A$ only means that $B-I$ belongs to the annihilator of $A$ in the ring $M_{n\times n}$.

Answer 2

It's false. $A$ could be the zero matrix, then $B$ could be absolutely any matrix.