Can (I-A)^{-1} Be Expressed as a Series When A^4 = 0?

[delgeezee]

Let A be a square matrix,

a) show that $$(I-A)^{-1}= I + A + A^2 + A^3 if A^4 = 0$$

b) show that $$(I-A)^{-1}= I + A + A^2+...+A^n $$ if $$
A^{n+1}= 0$$

 

[Amer]

Re: polynomial problem proof?

B is the inverse of A iff $$AB = BA = I$$
so
try
$$(I -A )( I + A + A^2 + A^3) = I-A + A - A^2 + A^2 - A^3 + A^3 - A^4 = I - A^4 = I$$