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Homework Help: Linear Algebra Inverse Generalization

  1. May 19, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that if [tex]T[/tex] is a square (matrix) and if [tex]T^4[/tex] is the zero matrix then [tex](I-T)^{-1} = I+T+T^2+T^3[/tex]. 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. jcsd
  3. May 19, 2010 #2

    gabbagabbahey

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    Well, what is [itex](I+T+T^2+T^3)(I-T)[/itex]?
     
  4. May 19, 2010 #3
    Is it...[tex]I[/tex]?
     
  5. May 19, 2010 #4

    lanedance

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    confirm it by multplying out. matrix addition & multiplication are distrubutive.
     
  6. May 19, 2010 #5
    Yeah, I know. It was a joke; of course its I. Everything in Linear is I.
     
  7. May 19, 2010 #6

    gabbagabbahey

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

    Now, can you generalize this statement?
     
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