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

Question

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

I don't know how to prove that. please help.

Answer 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 $(A+B)^{-1}=\begin{pmatrix}2&0\\0&2\end{pmatrix}^{-1}=\begin{pmatrix}\frac12&0\\0&\frac12\end{pmatrix}\neq \begin{pmatrix}1&0\\0&1\end{pmatrix}+\begin{pmatrix}1&0\\0&1\end{pmatrix}$

Answer 2

Why go to matrices when it doesn't even apply to scalars?

Let $A = 1$ and $B = 2$ so $(A+B)^{-1} = \frac{1}{3}$. BUT $$A^{-1} + B^{-1} = \frac{1}{1} + \frac{1}{2} = \frac{3}{2} \neq (A+B)^{-1} $$