I have some difficulty proving the bolded part of the exercise:

If A is an nxn matrix establish the identity

I_{n}-A^{k+1}=(I_{n}-A)(I_{n}+A+A^{2}+...+A^{k}).

Deduce that if some power of A is the zero matrix then I_{n}-A is invertible.

Suppose now that

A=2 2 -1 -1

-1 0 0 0

-1 -1 1 0

0 1 -1 1

Compute the powers (I_{n}-A)^{i}for i=1,2,3,4 and, by considering

A=I_{4}-(I_{4}-A),prove that A is invertibleand determine A^{-1}.

I would be grateful for any help you are able to provide...