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.

Solution 1:

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.

Solution 2:

$$(A+B)(A^{-1}+B^{-1})=2I+AB^{-1}+BA^{-1}$$ so your statement is true if and only if $I+AB^{-1}+BA^{-1}=0$ (which is of course not always true, and even usually wrong).