Originally Posted by

**ginarific** Hello!

This is one problem I've been tackling for a while:

If A, B, and A+B are all invertible matrices of order "n", prove that $\displaystyle A^{-1} + B^{-1}$ is invertible and that:

$\displaystyle (A^{-1}+B^{-1})^{-1} = A(A+B)^{-1}B = B(A+B)^{-1}A$

First, is there a generic formula for $\displaystyle A^{-1} + B^{-1}$? (If so, our instructor never taught us one). I'm also assuming that we can't assume AB is commutable, because that could make everything so much easier. ^_^;

Anyway. Any helpful hints might help. Basically what I've tried to do so far is equate the "middle" and RHS statements by multiplying both sides by inverses to isolate $\displaystyle (A+B)^{-1}$, but I don't really see how that 'proves' anything ...