sum of matrix inverse problem

Question

Recently, when I was reading matrix analysis, a formula confused me a lot:

If $A+B$ is nonsingular, then the following is true,

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

I tested some random samples on Matlab, it seems to be numerical correct. I wonder how to derive this formula?

Answer 1

Here is one way:

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

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

Subtracting the equations gives: $B (A+B)^{-1} A = A(A+B)^{-1} B$.