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

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

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