Linear Algebra Matrix Inverse Proof

[RoKr93]

Homework Statement

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.

Homework Equations

n/a

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.

 

[vela]

Homework Statement

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.

Homework Equations

n/a

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.

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?

 

[RoKr93]

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

 

[vela]

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

 

[RoKr93]

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

Thank you!