Prove or disprove that AB=AC $\implies$ B=C I proved it as follows but I'm not so sure about it.

A, B and C are square matrices of the same order.

Assume  $ B \neq C $

$$ AB \neq AC$$
$$ B \neq C \implies AB \neq AC$$
$$ \neg ( AB \neq AC) \implies \neg ( B \neq C ) $$
$$AB =AC \implies B=C $$
What if $A$ is zero? ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
If A be non-zero,then it is true.