Prove the following: If $A,B,C$ are all $n \times n$ matrices where $AB=CA$ and $A$ is invertible, then $B=C$
Here is my attempt at the solution, but I'm stuck
Let $D$ be the $n \times n$ inverse matrix of $A$, then $AD = I = DA$. Then,
$B = (I)B = (DA)B= D(AB) = A(CA)$
After that, I'm stuck. What should I do next?