Proving a matrix identity

Question

Let $M \in \mathbb{R}^{n\times n}$ with $\|M\| < 1$. Show

$$(I - M)^{-1} = I + M(I - M)^{-1}.$$

How can I do this? I tried starting with the equality

$$(I - M)(I - M)^{-1} = I, $$

Then I multiplied each side by $M$ to get

$$M(I - M)(I - M)^{-1} = M,$$

but I got nowhere from here

Answer 1

Just to make a connection with your earlier post:

• From there you know that $I-M$ is invertible, otherwise you had $||M|| \geq 1$.

Now, multiplying both sides of the given equation from the right by $I-M$ gives you

$$(I - M)^{-1} = I + M(I - M)^{-1}\stackrel{|\cdot (I-M)}{\Longrightarrow}I = (I-M) + M$$

Hence, you have also $$(I - M)^{-1} = I + M(I - M)^{-1}{\stackrel{|\cdot (I-M)^{-1}}{\Longleftarrow}}I = (I-M) + M$$

Answer 2

Use the fact that$$\bigl(\operatorname{Id}+M(\operatorname{Id}-M)^{-1}\bigr)(\operatorname{Id}-M)=\operatorname{Id}-M+M=\operatorname{Id}.$$

Answer 3

Since $||M||<1$, the series $$ \sum_{k\geq 0}M^k $$ makes sense, and of course $$ \sum_{k=0}M^k=I+M(\sum_{k\geq 0}M^k) $$