I have this equality

$$(A^{-1} + B^{-1})^{-1}=A(A+B)^{-1}B$$ where $A$ and $B$ are square symmetric matrices.I have done many test of R and Matlab that show that this holds, however I do not know how to prove it.

**Answer**

Assuming $A$, $B$, $A+B$, and $A^{-1}+B^{-1}$ are all invertible, note that

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

and then invert both sides, *QED*.

Symmetry is unnecessary for this to hold.

**Attribution***Source : Link , Question Author : Wis , Answer Author : whuber*