I'm trying to show that if is invertible then:
I have no idea where to start...?
Yes that's an interesting exercise:
Let be a ring, and two elements in .
Prove that if is left invertible (resp. right invertible), then is left invertible (resp. right invertible).
Here, is invertible, and the inverse of is given, but you have to verify it is correct.
So just multiply by .
is a left inverse of , therefore it's the inverse of .