Linear Algebra Matrix Equation Problem

Question

Let $A\in \mathbb R^{n×n}$ , $B\in \mathbb R^{n×n}$, Show that if $I + AB$ is nonsingular then

$\left ( I+AB \right )^{-1}=I-A\left ( I+BA \right )^{-1}B$

Answer 1

Hint:-

A matrix A is non-singular iff it is invertible.

Since $I+AB$ is nonsingular implies, it is invertible.

Using $series$ $expansion$ $$(I+BA)^{-1}=I-BA+(BA)(BA)-(BA)(BA)(BA)+\cdots$$ $$A(I+BA)^{-1}B=AB-(AB)(AB)+(AB)(AB)(AB)+(AB)(AB)(AB)(AB)+\dots$$ $$A(I+BA)^{-1}B=-(I+AB)^{-1}+I$$ $$(I+AB)^{-1}=I-A(I+BA)^{-1}B$$