Is there a proof for the following identity?
$\left(A^{-1}+B^{-1}\right)^{-1}=A(A+B)^{-1}B$
Is there a proof for the following identity?
$\left(A^{-1}+B^{-1}\right)^{-1}=A(A+B)^{-1}B$
$ \left( B^{-1}A + I\right) = \left(I + B^{-1}A\right) $
$ B^{-1} \left( A +B \right) = \left(A^{-1} + B^{-1}\right) A $
$ \left(A^{-1} + B^{-1}\right)^{-1} B^{-1} = A \left( A +B \right)^{-1}$
$ \left(A^{-1} + B^{-1}\right)^{-1} = A \left( A +B \right)^{-1} B$
\begin{equation} (A^{-1} + B^{-1})^{-1} = \left(B^{-1}(A + B) A^{-1}\right)^{-1} \end{equation}
Since $(CD)^{-1} = D^{-1} C^{-1}$, we have
\begin{equation} \left(B^{-1}(A + B) A^{-1}\right)^{-1} = A(A+B)^{-1}B \end{equation}
Write the matrices as $(A^{-1})^{-1}$ and repeatedly use $(M N)^{-1}=N^{-1}M^{-1}$.