Homework Help: Linear Algebra Matrix Inverse Proof

1. Feb 13, 2013

RoKr93

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
n/a

3. The attempt at a solution
I thought I'd want to use the fact that the multiplication of a matrix and its inverse is equal to I. So I started with (I-A)*(I + A + A^2 + A^3) = I. But that doesn't seem like the right direction...I'm not sure where to go from there.

2. Feb 13, 2013

vela

Staff Emeritus
What exactly do you mean you started with (I-A)*(I + A + A^2 + A^3) = I? Did you show the lefthand side equals the righthand side, or did you simply assert it?

3. Feb 13, 2013

RoKr93

I asserted it and wanted to start the proof from there, ie eventually get the same value on each side.

4. Feb 13, 2013

vela

Staff Emeritus
Just multiply (I-A)*(I + A + A^2 + A^3) out. What do you get?

5. Feb 13, 2013

RoKr93

Oh...wow. Guess it's been staring me in the face this whole time.

Thank you!