If $A$ and $B$ are $n^{\text{th}}$ order square matrices and $I-AB$ is invertible then what we can say about invertibility of $AB$?

Question

I think $AB$ is not necessary to be invertible.

Please explain by giving some examples. Original question is- TRUE/FALSE IF I-AB is invertible then AB is invertible.

Answer 1

$I-AB$ is invertible iff $1$ is not an eigenvalue of $AB$.

$\hphantom{I-{}}AB$ is invertible iff $0$ is not an eigenvalue of $AB$.

These two statements are independent.