Why is the inverse of a sum of matrices not the sum of their inverses? Suppose $A + B$ is invertible, then is it true that $(A + B)^{-1} = A^{-1} + B^{-1}$?
I know the answer is no, but don't get why.

Only for completeness, there are matrices $A$ and $B$ such as
$$(A+B)^{-1}=A^{-1}+B^{-1},$$
take, for instance,
$$A=\begin{bmatrix}
 -\frac{1}{2} & -\frac{\sqrt{3}}{2}  \\ 
 \frac{\sqrt{3}}{2} & -\frac{1}{2} 
\end{bmatrix}$$
and B the identity matrix. It is possible to build examples $A_n$ and $B_n$ when the dimension $n$ is even, just take $A_n$ as the diagonal block matrix with $A$ in the diagonal entries and $B_n$ the identity matrix. But there are no examples in odd dimensions!
