Answer
$B=C$
This is not true in general when A is not invertible.
Work Step by Step
Let $AB=AC$ where $B$ and $C$ are $n$$\times$$p$ matrices and $A$ is invertible.
$AB=AC$
$A^{-1}AB=A^{-1}AC$
$IB=IC$
$B=C$
Let $A=\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$, $B=\begin{bmatrix}2&0\\0&0\\\end{bmatrix}$, and $C=\begin{bmatrix}2&0\\0&1\\\end{bmatrix}$. A is not invertible because $detA=0$.
$AB=\begin{bmatrix}1&0\\0&0\\\end{bmatrix}\begin{bmatrix}2&0\\0&0\\\end{bmatrix}=\begin{bmatrix}2&0\\0&0\\\end{bmatrix}$
$AC=\begin{bmatrix}1&0\\0&0\\\end{bmatrix}\begin{bmatrix}2&0\\0&1\\\end{bmatrix}=\begin{bmatrix}2&0\\0&0\\\end{bmatrix}$
$AB=AC$ but $B$$\ne$$C$ so this is not true in general when $A$ is not invertible.
