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Homework Help: Skew-symmetric matrices problem ?

  1. Jul 19, 2007 #1
    1. The problem statement, all variables and given/known data

    Give an example of two skew-symmetric matrices. Show explicitly that they display the property of skew-symmetry, ie, AB = -BA

    2. Relevant equations

    3. The attempt at a solution

    transpose of (AB) = BA
    I just can show that AB=BA but can't show AB=-BA .
    Is it (-A)(-B)=AB ?
  2. jcsd
  3. Jul 20, 2007 #2


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    The defining property of a skew-symmetric matrix is transpose(A)=-A. So yes, transpose(AB)=(-B)*(-A)=BA. Beyond that I simply don't follow you. A=[[0,1],[-1,0]]. B=transpose(A)=[[0,-1],[1,0]]. AB=1=BA. AB is not equal to -BA. AB=-BA is property of anticommuting matrices.
    Last edited: Jul 20, 2007
  4. Jul 20, 2007 #3
    Since AB=BA, AB can't be -BA unless AB = 0
  5. Jul 20, 2007 #4


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    Nonsense. Multiplication of matrices is NOT in general commutative.
  6. Jul 20, 2007 #5
    I think i have misunderstood the questions. My lecturer say the question is we multiply any 2 matrices to get a skew-symmetry matrix AB . Then show that AB=-BA . But i simply can't show it .
    I even don't know how to give 2 matrices where the product of these 2 matrices is skew-symmetry matrix .
    I know i can use try but i think that is not a good and standard technique .
    Anybody has any idea on this question?
  7. Jul 20, 2007 #6


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    A=[[-1,-1,0],[0,1,1],[-1,0,1]], B=[[0,1,0],[0,0,1],[1,0,0]].

    AB is skew symmetric. AB is not equal to -BA. No wonder you can't show it. Do you mean to add the assumption A and B are symmetric?
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