Matrix cancellation requires hypotheses. If $A$ is invertible and $AB=AC$, multiply on the left by $A^{-1}$ to obtain $B=C$. For singular $A$, cancellation can fail: with $A=\operatorname{diag}(1,0)$, $B=0$, and $C=\operatorname{diag}(0,1)$, one has $AB=AC=0$ but $B\ne C$.
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