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Math Help - Matrices- Nilpotent

  1. #1
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    Matrices- Nilpotent

    show that if N is a nilpotent nxn matrix then identity matix+ N is invertible.
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  2. #2
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    Quote Originally Posted by amm345 View Post
    show that if N is a nilpotent nxn matrix then identity matix+ N is invertible.
    Since N is a nilpotent square matrix, N^k = 0 for some positive integer k.

    If k is an odd positive number greater than 1 (if k=1, then it is trivial), then the required inverse is

     I - N + N^2 - N^3 + ,,,, + N^{k-1}.

    To verify this,
    (I + N)(I - N + N^2 - N^3 + ,,,, + N^{k-1})
    = I - N + N^2 + ,,,, + N^{k-1} +N - N^2 + ...- N^{k-1} + N^{k}
    = I , since N^k = 0.

    If k is an even positive number, then the required inverse is
     I - N + N^2 - N^3 +,,,, - N^{k-1}. You can verify this in the same manner as the above.
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