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Linear Algebra Matrix Inverse Proof

  1. Feb 13, 2013 #1
    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. jcsd
  3. Feb 13, 2013 #2

    vela

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    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?
     
  4. Feb 13, 2013 #3
    I asserted it and wanted to start the proof from there, ie eventually get the same value on each side.
     
  5. Feb 13, 2013 #4

    vela

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    Just multiply (I-A)*(I + A + A^2 + A^3) out. What do you get?
     
  6. Feb 13, 2013 #5
    Oh...wow. Guess it's been staring me in the face this whole time.

    Thank you!
     
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