I have some difficulty proving the bolded part of the exercise:
If A is an nxn matrix establish the identity
Deduce that if some power of A is the zero matrix then In-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 (In-A)i for i=1,2,3,4 and, by considering
A=I4-(I4-A), prove that A is invertible and determine A-1.
I would be grateful for any help you are able to provide...