Factoring a series of Matricies

Question

I have a graduate physics text which has an arbitrary matrix $M$. I am given the following $M^{0} + M^{1} + M^{2}...M^{n-1}$ and then it says this is equal to $(M^{n}-I)(M-I)^{-1}$, where $I$ is the identity matrix. I dont not see how this follows, IF anyone can show me I would appreciate it.

It looks like some sort of power series of Matrices, but I never could fill in the gap.

Thanks, Ben

Answer 1

$$(M-I)(I+M+\cdots+M^{n-1})$$ $$(I+M+\cdots+M^{n-1})(M-I)$$ $$=(M+M^2+\cdots+M^{n-1}+M^n)$$ $$-(I+M+\cdots+M^{n-2}+M^{n-1})$$ $$=M^n-I$$

If $1$ is not the eigenvalue of $M$, that is to say $M-I$ is invertible, then $$I+M+\cdots+M^{n-1}=(M-I)^{-1}(M^n-I)=(M^n-I)(M-I)^{-1}$$

This approach is similar to the geometric progression。