How to show two matrices commute?

Question

How do I show that $(I_n+M)$ and $(I_n-M)^{-1}$ commute where $I_n$ is the $n\times n$ identity matrix and $M$ is an $n\times n$ matrix.?

I have been trying to figure this out for ages, I think I must be missing something simple, any help would be much appreciated.

Answer 1

You have $$ I=(I-M)(I-M)^{-1}=(I-M)^{-1}-M(I-M)^{-1}, $$ and $$ I=(I-M)^{-1}(I-M)=(I-M)^{-1}-(I-M)^{-1}M. $$ Comparing the two equalities, you get $$ (I-M)^{-1}M=M(I-M)^{-1}. $$

Answer 2

For brevity denote $I_n+M=A,$ and $I_n-M=B.$ Assuming that $B$ is invertible, we want to show that $AB^{-1}=B^{-1}A.$ To do so, consider that $A+B=2I_n,$ so multiplying both sides by on the right by $B^{-1},$ we get

$$B^{-1}(A+B)=B^{-1}2I_n=2B^{-1},$$ So $B^{-1}(A+B)=2B^{-1},$ a similar argument by multiplying on the left gives,

$$(A+B)B^{-1}=2I_nB^{-1}=2B^{-1},$$

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

$$B^{-1}A +I_n= AB^{-1}+I_n.$$

Subtracting $I_n$ from both sides yields the desired result.