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.

Let us look at the often-forgotten $1 \times 1$ matrices over $\mathbb{R}$, which is another name for the real numbers themselves. Then your statement translates to the statement that if $x,y$ are real numbers that aren't zero, then
$$ \frac{1}{x + y} = \frac{1}{x} + \frac{1}{y},$$
which is clearly wrong.
