# properties of invertible matrices $(A+B)^{-1} = (A^{−1})+(B^{−1})$

If $A,B$ and $A+B$ are all $n \times n$ invertible matrices. Prove that $(A^{−1})+(B^{−1})$ is equal to $(A+B)^{-1}$

• Usually it's difficult to prove things that are false. – Eric Wofsey Sep 10 '18 at 1:34
• You made my day @EricWofsey – Ahmad Bazzi Sep 10 '18 at 1:35
• Try to multiply $\,\left(A+B)\cdot(A^{-1}+B^{-1}\right)\,$ and see if you get the identity matrix. – dxiv Sep 10 '18 at 1:36
Let $A = 1$ and $B = 2$ so $(A+B)^{-1} = \frac{1}{3}$. BUT $$A^{-1} + B^{-1} = \frac{1}{1} + \frac{1}{2} = \frac{3}{2} \neq (A+B)^{-1}$$
This is wrong. For example: $A=B=\begin{pmatrix}1&0\\0&1\end{pmatrix}$
Then $A^{-1}=B^{-1}=\begin{pmatrix}1&0\\0&1\end{pmatrix}$
But $(A+B)^{-1}=\begin{pmatrix}2&0\\0&2\end{pmatrix}^{-1}=\begin{pmatrix}\frac12&0\\0&\frac12\end{pmatrix}\neq \begin{pmatrix}1&0\\0&1\end{pmatrix}+\begin{pmatrix}1&0\\0&1\end{pmatrix}$