Question on matrices inverses

Question

Let $A\in M_{m\times n}(\mathbb C)$ and $B\in M_{n\times m}(\mathbb C)$ be two matrices. If $AB$ is invertible, does it mean that $A$ and $B$ are invertible? If so, then how do I prove that? Thanks in advance!

Answer 1

No.

To be invertible, a matrix has to be a square matrix which is not the case here.

If they are indeed square matrices, note that $\det(AB)=\det(A)\det(B)$. Try to prove it using this identity.