Let $A,B$ be $m\times m$ matrices such that $AB$ is invertible. Show $A,B$ invertible. [duplicate]

Okay, so I figured this out.

We have a left and right inverse for our matrices $B$ and $A$ respectively. Since $A,B$ both correspond to linear transformations from a space of dimension n to another of dimension n we know that they are functions which possess a right and left inverse respectively, and thus are onto and one-to-one respectively. Therefore they are both one-to-one and onto since having one property gives the other when we are dealing with vector spaces of finite and equal dimension. Since $A,B$ both correspond to invertible transformations we know they are invertible. Q.E.D.

$AB$ is invertible so $\det(AB)\neq 0$. We have $\det(AB)=\det(A)\det(B)$; since it is nonzero it follows that neither $\det(A)$ or $\det(B)$ are zero, so $A$ and $B$ are both invertible.