# Polynomial problem proof?

#### delgeezee

##### New member
Let A be a square matrix,

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

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

#### Amer

##### Active member
Re: polynomial problem proof?

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