Answer
$A=\left[\begin{array}{cc} 2 & 6\\ 0 & 2\end{array}\right],B=\left[\begin{array}{cc} 4 & 8\\ 4 & 6\end{array}\right]$ and $C=\left[\begin{array}{rr} 5 & -10\\ -5 & 10\end{array}\right]$ give us a counterexample for the statement.
Work Step by Step
Consider the following statement:
If $A,B$, and $C$ are matrices and $AC=BC$, then $A=B$.
We will now present a counterexample in order to show that the statement is false.
Consider the matrices $A=\left[\begin{array}{cc} 2 & 6\\ 0 & 2\end{array}\right],B=\left[\begin{array}{cc} 4 & 8\\ 4 & 6\end{array}\right]$ and $C=\left[\begin{array}{rr} 5 & -10\\ -5 & 10\end{array}\right]$.
Observe that
$$AC=\left[\begin{array}{cc} 2 & 6\\ 0 & 2\end{array}\right]\left[\begin{array}{rr} 5 & -10\\ -5 & 10\end{array}\right]=\left[\begin{array}{cc} -20 & 40\\ -10 & 20\end{array}\right]$$
and
$$BC=\left[\begin{array}{cc} 4 & 8\\ 4 & 6\end{array}\right]\left[\begin{array}{rr} 5 & -10\\ -5 & 10\end{array}\right]=\left[\begin{array}{cc} -20 & 40\\ -10 & 20\end{array}\right].$$
Hence, we have that $AC=BC$ and $A\neq B$ and therefore, the statement is false.
