# If A,B, and C are nxn matrices, does AB=AC mean B = C?

No. What if $A = 0$ ?
For example, $A = \left(\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}\right)$, $B = \left(\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right)$, $C = \left(\begin{matrix}5 & 6 \\ 7 & 8\end{matrix}\right)$
Then $A B = A C = \left(\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}\right)$, but $B \ne C$