TITLE: Inverse of $A^{-1}+B^{-1}$
QUESTION [18 upvotes]: If $A, B$ and $A + B$ are all $n × n$ invertible matrices. Prove that $A^{−1} + B^{−1}$ is invertible and the inverse is $A(A + B) ^{−1}B$.
I am afraid I am really stuck on this one, and I haven't really tried much because I don't know what to try.
Thanks for all the help guys, I understand now.

REPLY [11 votes]: Here's a concise proof:
$$
\begin{align}
A^{-1} + B^{-1} &= B^{-1} + A^{-1} \\
&= B^{-1}AA^{-1} + B^{-1}BA^{-1}\\
&= B^{-1}(A + B)A^{-1} \\
\end{align}
$$
So $A^{-1} + B^{-1}$ is the product of invertible matrices and thus is invertible, with inverse equal to
$$
\begin{align}
(A^{-1} + B^{-1})^{-1} &= (B^{-1}(A + B)A^{-1})^{-1} \\
&= A(A + B)^{-1}B \\
\end{align}
$$