If for two suitable matrices $B$ and $C$, $AA^TB=AA^TC$ can we conclude that $A^TB=A^TC$? 
Let $A$ be a $m\times n$ real matrix. If for two suitable matrices $B$ and $C$,
$$AA^TB=AA^TC$$
can we conclude that $A^TB=A^TC$?

I know $AA^T$ is a symmetric matrix. Every real symmetric matrix is diagonalizable. But I am not sure how to use this information.
Thanks for reading and helping me out with hints.

$AA^TD=0\Rightarrow (A^TD)^T(A^TD)=D^TAA^TD=0\Rightarrow A^TD=0$. Now put $D=B-C$.
