# Prove that $A(A+B)^{-1}B=B(A+B)^{-1}A$

Given $A$, $B$ are two square matrices, $A+B$ is non-singular, prove above state. This is one part of my algebra mid term, of course, I failed, I have no idea where I should start. One thing appeared in my mind during the exam is $A(I+A)=(I+A)A$, $(I+A)$ is non-singular, how to represent A+B under that form... That maybe a wrong idea.

Observe that $$\text{LHS}=A(A+B)^{-1}B=A(A+B)^{-1}(A+B-A)=A-A(A+B)^{-1}A$$ and $$\text{RHS}=B(A+B)^{-1}A=(A+B-A)(A+B)^{-1}A=A-A(A+B)^{-1}A.$$