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Math Help - Nilpotent matrix Invertible matrix

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Nilpotent matrix Invertible matrix

    Let A in M_m(F), be a Nilpotent matrix.

    Prove the following:
    1. A is a non-invertible matrix.

    2. Order of nilpotentence of A small or equals to m.

    3. I_n + A + A^2 +... + A^(k-1) is invertible when k is nilpotentence order of A.

    4. If B is in M_n(F), a matrix. Is A+B must be Nilpotent matrix?

    5. If B is in M_n(F), a matrix. Is A*B must be Nilpotent matrix?

    6. Find B in M_5(R) Nilpotent matrix with order of 5.


    Last edited by Also sprach Zarathustra; January 6th 2010 at 11:06 AM.
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  2. #2
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Let A in M_m(F), be a Nilpotent matrix.

    Prove the following:
    1. A is a invertible matrix.

    This is false, of course: if a matrix mis nilpotent then its determinant is zero and thus it isn't invertible.

    Tonio



    2. Order of nilpotentence of A small or equals to m.

    3. I_n + A + A^2 +... + A^(k-1) is invertible when k is nilpotentence order of A.

    4. If B is in M_n(F), a matrix. Is A+B must be Nilpotent matrix?

    5. If B is in M_n(F), a matrix. Is A*B must be Nilpotent matrix?

    6. Find B in M_5(R) Nilpotent matrix with order of 5.

    .
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Yes! it's my mistake. A is a non-invertible matrix.
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