Prove $A(A+B)^{-1}B = B(A+B)^{-1}A$ where $(A+B)^{-1}$ and $A$ and $B$ are $n\times n$ matricies.

Question

Hi I have been working on this problem for the longest time.

Prove:

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

We know that A & B exist in real space, and that they are also N x N matrices. It is also given that $(A + B)^{-1}$ is true. We do not know anything about A or B however except that their sum is invertible.

I was going to use the SMW Identity but the issue is that you need to assume that either A or B is invertible which we cannot assume in this case.

Thanks.

Answer 1

Proceed as follows: $$ A(A+B)^{-1}B =\\ [(A + B) - B](A+B)^{-1}B =\\ (A + B)(A+B)^{-1}B - B(A+B)^{-1}B=\\ B - B(A+B)^{-1}B =\\ B - B(A+B)^{-1}[(A + B) - A] =\\ B - [B(A+B)^{-1}(A + B) - B(A+B)^{-1}A]=\\ B - [B - B(A+B)^{-1}A]=\\ B(A+B)^{-1}A $$