Existence of Matrix inverses depending on the existence of the inverse of the others.. Let $A_{m\times n}$ and $B_{n\times m}$ be two matrices with real entries. Prove that $I-AB$ is invertible iff $I-BA$ is invertible.

Hint: Assume $m<n$, use the fact that $p_{BA}(t)=t^{n-m}p_{AB}(t)$, where $p_{AB}$ is the characteristic polynomial of $AB$. Similar for $m>n$.
