Matrix equation: $A−A(A + B)^{−1}A = B−B(A + B)^{−1}B$

$$A−A(A + B)^{−1}A = B−B(A + B)^{−1}B$$

A+B is nonsingular

So i got this equation i have to show that both sides are equal but only A + B is nonsingular A and B are not. I got to how solve it when A and B are also nonsingular but i realised that there aren't. So now i dont get how to solve it. Can someone give me a hint on how to do it?

• What do you want exactly ? Solve for what ? What is the context ? What did you try ? – Nicolas FRANCOIS Nov 25 '18 at 12:14
• Nicolas FRANCOIS, as i said i have to show both sides of the equation are equal. A and B are both nxn matrices. – Danielvanheuven Nov 25 '18 at 14:43

Since $$A+B$$ is non-singular we have that $$(A+B)^{-1}(A+B)=I\implies A(A+B)^{-1}A+A(A+B)^{-1}B=A.$$ and similarily $$(A+B)(A+B)^{-1}=I\implies A(A+B)^{-1}B+B(A+B)^{-1}B=B.$$ Thus on subtracting the two we get: $$A(A+B)^{-1}A-B(A+B)^{-1}B=A-B.$$ Rearrange this to get your answer.