TITLE: How to prove that $A^{-1} + B^{-1}$ is invertible given the conditions
QUESTION [6 upvotes]: If $A$ and $B$ be two invertible $n \times n$ real matrices and $A + B$ is invertible, how to prove that $A^{-1} + B^{-1}$ is also invertible?

REPLY [13 votes]: Let $C=A^{-1}+B^{-1}$ then $C=A^{-1}(I+AB^{-1})=A^{-1}(B+A)B^{-1}$, So $C$ is invertible as the product of three invertible matrices.