For two n$\times$n matrices A and B, if AB=I then BA=I?

Question

In a book I'm reading, at some point it's said because we have found a n$\times$n matrix A so that AB=I (B being a n$\times$n matrix) then B is invertible and A is it's inverse, is it true and is there actually a proof for it? because I know for A to be inverse of B, BA=I should also be true.

Answer 1

You can see this from the determinant of the matrices. AB=I means det(A)*det(B)=1 meaning that det(A) and det(B) are not zero meaning that they are both invertible.