Linear Algebra Inverse Generalization

[gabriels-horn]

Homework Statement

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.

The Attempt at a Solution

To be honest I don't even know where to begin. Supposedly this is a simple generalization.

 

[gabbagabbahey]

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

 

[gabriels-horn]

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

Is it...$$I$$?

 

[lanedance]

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

 

[gabriels-horn]

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

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

 

[gabbagabbahey]

Is it...$$I$$?

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?