If $B$ the inverse matrix of $A^2$ show that the inverse of $A$ is $AB$ How do I continue from $A(AB) = (BA)A = I$ and how can we justify it if we don't know that $AB=BA$?
EDIT: Also, how can we prove that if $AB=I$ then $ BA = I$? This is seperate from the question above but I felt it didn't need a new post.

$A(AB)=(AA)B=A^2 B=I$ and $(BA)A=B(AA)=BA^2=I$
