How can I prove $A − A(A + B)^{−1}A = B − B(A + B)^{−1}B$ for matrices $A$ and $B$?

Question

The matrix cookbook (page 16) offers this amazing result:

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

This seems to be too unbelievable to be true and I can't seem to prove it. Can anyone verify this equation/offer proof?

Answer 1

\begin{align} A - A(A+B)^{-1}A & = A(A+B)^{-1}(A+B) - A(A+B)^{-1}A \\ &= A(A+B)^{-1}(A+B - A)\\ &= 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 \end{align}

Answer 2

You may simply put $X=A+B$ and show that \begin{aligned} A-AX^{-1}A &=(X-B)-(X-B)X^{-1}(X-B)\\ &=(X-B)-(X-2B+BX^{-1}B)\\ &=B-BX^{-1}B. \end{aligned}