# Are there non-square matrices that are both left and right invertible?

I am aware that invertible square matrices are left invertible and right invertible, and that the left and right inverses are equal. However, I was wondering whether exists a non square $m\times n$ matrice $A$, so that exist both:

1. An $n\times m$ matrice $B$ so that $AB = I_m$
2. An $n\times m$ matrice $C$ so that $CA = I_n$

I just couldn't think of an example nor of a proof that these two conditions provide that A is necessarily square.

• If $A$ and $B$ are matrices, then each is a matrix, not matrice. – Omnomnomnom Apr 4 '17 at 19:34
• Note that $\operatorname{rank}(AB) \leq \operatorname{rank}(A) \leq \min\{m,n\}$ and $\operatorname{rank}(CA) \leq \operatorname{rank}(A) \leq \min\{m,n\}$. $AB$ and $CA$ cannot both have full rank. – Omnomnomnom Apr 4 '17 at 19:36

## 1 Answer

Suppose you have an $n\times m$ matrix $A$ with $n\neq m$.

If $n<m$ then $rank(BA)\leq rank(A)\leq n$ and so $BA\neq I_m$ for all $B$.

If $m<n$ then $rank(AB)\leq rank(A)\leq m$ and so $AB\neq I_n$ for all $B$.

• Awesome, thanks. – Noa Apr 6 '17 at 13:15