# Homework Help: Linear Algebra Inverse Generalization

1. May 19, 2010

### gabriels-horn

1. The problem statement, all variables and given/known data
Show that if $$T$$ is a square (matrix) and if $$T^4$$ is the zero matrix then $$(I-T)^{-1} = I+T+T^2+T^3$$. Generalize.

3. The attempt at a solution
To be honest I don't even know where to begin. Supposedly this is a simple generalization.

2. May 19, 2010

### gabbagabbahey

Well, what is $(I+T+T^2+T^3)(I-T)$?

3. May 19, 2010

### gabriels-horn

Is it...$$I$$?

4. May 19, 2010

### lanedance

confirm it by multplying out. matrix addition & multiplication are distrubutive.

5. May 19, 2010

### gabriels-horn

Yeah, I know. It was a joke; of course its I. Everything in Linear is I.

6. May 19, 2010

### gabbagabbahey

Multiplying it out should give you $I-T^4$. So, if $T^4=0$,, then you get the identity matrix and hence $(1+T+T^2+T^3+T^4)=(I-T)^{-1}$.

Now, can you generalize this statement?