When do addition and inversion of matrices commute? Can you guys help me out with the following problem:
Problem: Find conditions that the matrices $A$ and $B$ have to satisfy in order for the following to be valid: $(A+B)^{-1} = A^{-1} + B^{-1}$.
My solution. The first condition is that $A$ and $B$ have to have the same dimensions. Then by expanding the two equalities: $(A+B) \cdot (A^{-1} + B^{-1})=I $ and $(A^{-1} + B^{-1}) \cdot (A+B)=I$, I deduced the second condition: $A$ and $B$ have both to be invertible, and third condition is: $A^{-1} \cdot B=B \cdot A^{-1}$. Is my answer correct? 

For third condition you have that:
$(A+B)(A+B)^{-1}=(A+B)(A^{-1}+B^{-1})$
$(A+B)^{-1}(A+B)=(A^{-1}+B^{-1})(A+B)$
$\Rightarrow (A+B)(A^{-1}+B^{-1})=(A^{-1}+B^{-1})(A+B)\Rightarrow $
$\Rightarrow AA^{-1}+AB^{-1}+BA^{-1}+BB^{-1}=A^{-1}A+A^{-1}B+B^{-1}A+B^{-1}B \Rightarrow$
$\Rightarrow AB^{-1}+BA^{-1} =A^{-1}B+B^{-1}A$
