Show that $A^{-1} + B^{-1}$ is invertible when $A,B$ and $A+B$ are invertible

Question

I have the following issue: $A,B\in\mathbb C^{n\times n}$ invertible, such that also $A + B$ is invertible. How is it shown that $A^{-1} + B^{-1}$ is invertible?

Answer 1

$$ A^{-1}+B^{-1} = A^{-1}(A+B)B^{-1} $$ By the way: (Spanish) demonstración --> (English) proof. ;-)

Answer 2

Of course you know that a square matrix has a two sided inverse if you have found an inverse from one side.

Answer 3

Here is a more pedantic approach to @amsmath's slick approach:

Suppose we want to solve $(A^{-1} + B^{-1}) x = A^{-1}x + B^{-1} x = y$.

Then $Ay=x + AB^{-1} x $, and letting $x'=B^{-1} x$ we get $Ay = B x' + A x' = (A+B)x'$ and so $x'= (A+B)^{-1} Ay$ and finally $x=Bx' = B(A+B)^{-1} Ay$.

Hence $(A^{-1} + B^{-1})^{-1} = B(A+B)^{-1} A$.