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Math Help - Matrix Problem (Proving Invertibility)

  1. #1
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    Matrix Problem (Proving Invertibility)

    In this exercise A is a square matrix.
    1) Show that if A^2 = 0 then (A - I) is invertible.
    2) Show that if A^3 = 0 then (A - I) is invertible.
    3) Show that if A^4 = 0 then (A - I) is invertible.
    4) Assume that there is a natural number k such that A^k = 0. Prove that (A - I)
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  2. #2
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    When I first did this question, consider the inverse of (x-1) for a real number |x| <1 and see if you can construct an inverse
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  3. #3
    Super Member Gamma's Avatar
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    Expanding on previous help

    Formula that might be of use: remember the geometric series formula.
    \frac{1}{1-x}=\sum_{n=0}^{\infty}x^n for |x|<1. But this is just to ensure your series converges. In this case your matrix is nilpotent, so your infinite series in fact is not quite so infinite.
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