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Proof regarding skew symmetric matrices

  1. Oct 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that if A is skew symmetric, then Ak is skew symmetric for any positive odd integer k.

    2. Relevant equations

    3. The attempt at a solution

    Wow, I have no idea how to prove this. I'm guessing there's going to be induction involved. I know that the base case of k = 1 is true, because A1 = A, where A is skew symmetric. But after that, I don't know what to do at all.

    Any suggestions on where to go from here?
  2. jcsd
  3. Oct 14, 2009 #2


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    Skew-symmetric means A^(T)=(-A). (T=transpose). Does that help? Can you use that to show A^3 is skew-symmetric?
  4. Oct 14, 2009 #3
    A &= -A^T\\
    A^3 &= (-A^T)^3\\
    &= (-A^T)(-A^T)(-A^T)\\
    &= (A^2)^T(-A^T)\\
    &= (-AA^2)^T\\
    &= (-A^3)^T

    so A3 is skew symmetric.

    Thanks again for your help Dick. Now, how can I go about proving this for the rest of the positive odd integers. Will induction work? I'm trying to think of a way to do it, but I'm stuck on the "odd" integers part.
  5. Oct 14, 2009 #4


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    Sure, induction will work. But that's probably overkill. (A^n)^T=(A^T)^n. Just factor out the (-1)^n.
  6. Oct 14, 2009 #5
    OH...haha thanks very much. I tend to over analyze sometimes...
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