The question says:
If matrix $A$ is invertible and $A + B = AB$, prove that matrix $B$ is invertible and $A^{-1} + B^{-1} = I$
Firstly, I was thinking that we can only prove that matrix $B$ is a square matrix because matrix $A$ is a square matrix. But I can't think of a way to prove that it's invertible as well.
For the rest of the question, can I write $A^{-1} + B^{-1}$ as $(A + B)^{-1}$? I don't think it's correct.
Thanks for any help in advance.
First we use the equation given as
$$A^{-1}(A+B)=A^{-1}AB$$
which implies $$I+A^{-1}B=B.$$
Then to find the inverse of $B$ we start by assuming that $B$ is invertible which leads to
$$(I+A^{-1}B)B^{-1}=BB^{-1}=I.$$
This gives
$$B^{-1}+A^{-1}=I$$
and then we can write $$B^{-1}=I-A^{-1}$$ which we can check satisfies:
$$(I-A^{-1})B=B-A^{-1}B=I$$