$AB = C$, prove if $C$ is invertible, then $A$ and $B$

Question

Let $A, B, C$ be $n \times n$ matrices and assume $AB = C$. Prove if $C$ is invertible, then $A$ and $B$ are invertible and specify inverses.

Answer 1

Hint: Let $D$ be the inverse of $C$. Then $CD=DC=I$. Now multiply $AB=C$ by $D$ separately on the right and on the left.

If you don't need to give the inverses, just compute the determinant on both sides.