linear algebra - 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.