Let 2 $v \times v$ invertible matrices $A,B$ such that $A+B$ to be invertible.Show that the matrix $A^{-1}+B^{-1}$ is invertible and $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B=B(A+B)^{-1}A$
My effort :
We know that $A+B$ is invertible.So $$(A+B)(A+B)^{-1}=I$$
I have to show that :
\begin{align*} AA^{-1}&=I \\ A^{-1}+B^{-1})(A^{-1}+B^{-1})^{-1}&=I\\ \\ \end{align*}
\begin{align*} A^{-1}+B^{-1})(A^{-1}+B^{-1})^{-1}&= (A^{-1}+B^{-1}) \left[ B(A+B)^{-1}A \right]\\ &= A^{-1} \left[ B(A+B)^{-1}A \right] +B^{-1} \left[ B(A+B)^{-1}A \right]\\ &= B(A+B)^{-1}AA^{-1} + B^{-1}B(A+B)^{-1}A \\ &= B(A+B)^{-1} +A(A+B)^{-1}\\ &= (A+B)^{-1}(A+B) \\ &=I \end{align*}
Am I correct ? Is there another way ?
