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Matrices Inverse properties

  1. Feb 15, 2007 #1
    1. The problem statement, all variables and given/known data

    Determine which of the formulas hold for all invertible nxn matrices A and B

    1. (A+A^-1)^7 = A^7+A^-7
    2. 2A is invertible
    3. (A+B)² = A² + B² + 2AB
    4. (ABA^-1)^6 = AB^6A^-1
    5. A + B is invertible
    6. ABA^-1 = B


    I know 2A should be invertible, and number 3 wrong, but what else?
    Thanks
     
  2. jcsd
  3. Feb 15, 2007 #2

    radou

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    Try to think of examples if a fact bothers you or if you can't proove it.
     
  4. Feb 16, 2007 #3

    HallsofIvy

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    For 1) and 3, write out the computation- for 1 you might want to look at (A+ A-1)2 first.

    For 5, if A is invertible, then so is B= -A.
     
  5. Feb 16, 2007 #4

    D H

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    Think of whether 1 and 5 are true for all invertible 1x1 matrices (i.e., the non-zero reals or non-zero complex numbers).

    For 4, expand the left-hand side.

    For 6, what are the conditions that a matrix B be self-similar with respect to A?
     
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