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Applied Linear Algebra problem

  1. Sep 5, 2011 #1
    the question:

    the matrix

    1 -1
    1 -1

    has the property that A2 = 0. Is it possible for a nonzero symmetric 2x2 matrix to have this property? Prove your answer.

    my work:

    for a 2x2 matrix A to be its own inverse, it has to have the form

    a b
    b a

    This squared is

    (a2 + b 2) (2ab)
    (2ab) (a2 + b2)

    (things in parenthesis are their own elements -- it wont save the spaces)

    Because there are no real numbers so that a2 + b2 = 0, there is no 2x2 symmetric matrix that has its square equal to the zero vector.

    edit: ^^ other than a = 0, and b = 0, which would be a 2x2 zero matrix -- something taken to account in the statement of the question

    Is this right? My book doesnt have a solution for this one
     
    Last edited: Sep 5, 2011
  2. jcsd
  3. Sep 5, 2011 #2
    How about

    [tex]\left( {\begin{array}{*{20}{c}}
    2 & 4 \\
    { - 1} & { - 2} \\
    \end{array}} \right)[/tex]

    Edit
    Sorry I thought you meant a nonsymmetric matrix. What do you mean by this?
     
    Last edited: Sep 5, 2011
  4. Sep 5, 2011 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    anonymity, yes, your analysis is correct.
     
  5. Sep 5, 2011 #4
    How did you write that matrix in physicsforum's latex?!

    And by nonzero they just mean it's not

    0 0
    0 0


    Thanks for responding hallsofivy
     
  6. Sep 6, 2011 #5

    Stephen Tashi

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    Science Advisor

    Click on the "quote" box in hallsofivy's post and look at how he wrote the matrix in your message composer window.
     
  7. Sep 6, 2011 #6
    Very clever. Thank you ^
     
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