How to prove $CAA^t=BAA^t$ implies $CA=BA$?

Question

Suppose $A$, $B$, $C$ are $n\times n$ matrices. $A'$ denotes the transpose of $A$. $CAA'=BAA'$. How to prove $CA=BA$?

Answer 1

The given equation $(C-B)AA'=0$ means that each row of the matrix $C-B$ is in the left kernel of $AA'$. The desired equation $(C-B)A=0$ means that each row of the matrix $C-B$ is in the left kernel of $A$. So, I want to show that the (left) kernel of the matrix $AA'$ is already the (left) kernel of the matrix $A$.

Suppose the vector $v$ is in the left kernel of $AA'$, i.e. $$vAA'=0$$ which implies $$vAA'v'=0=(vA)\cdot (vA)'=\|vA\|^2$$ (where $v'$ is the transpose of the vector $v$).

So, $vA$ is already the zero vector and therefore, $v$ is in the left kernel of $A$.