Matrix inversion is not additive ($(A+B{)}^{-1}\ne A^{-1}+B^{-1}$)

For $A=B=I$ of any positive size, $A+B=2I$ is invertible but $(A+B{)}^{-1}=I/2$, whereas $A^{-1}+B^{-1}=2I$. Thus even invertibility of all three matrices does not make inversion additive.