# Question on matrices inverses

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!

• How do you define the inverse of a $m×n$ matrix if $m\neq n$? – P. Siehr Nov 8 '17 at 8:35
• To clarify, my question was not meant to be offending in any way. Rather: Do you also think about pseudo inverses in your question. – P. Siehr Nov 8 '17 at 8:44
• No, $A$ and $B$ will not be invertible in general but you can prove that $A$ has right inverse and $B$ has left inverse. – A---B Nov 8 '17 at 8:50

## 1 Answer

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.

• Is there a solvation without using determinants? we've just started this chapter – Anonymus Nov 8 '17 at 9:12
• If $(AB)C = I$, then $A(BC)=I$, try to prove $B$ is invertible too for the context of square matrix. – Siong Thye Goh Nov 8 '17 at 9:19