[Math] How to prove that $AB$ is invertible if and only if $A$ is invertible

Question

Let $A$ be a matrix and $B$ an invertible matrix. Show that $AB$ is invertible if and only if $A$ is invertible.

I know how to do this using determinants, but how else could you prove this?

Answer 1

$A$ is invertible $\implies$ $AB$ is invertible: This is because $(AB)^{-1}=B^{-1}A^{-1}$.

$A$ is invertible $\impliedby$ $AB$ is invertible: use the proven implication ($\implies$) above, applied to the matrices $AB$ and $ABB^{-1}=A$, with the fact that, since $B$ is invertible, $B^{-1}$ is also invertible.