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Linear algebra problem

  1. Nov 11, 2008 #1
    1. The problem statement, all variables and given/known data
    An n x n matrix A is anti-symmetric if it satis fies the equation A^t = -A.
    Show that if n is odd and A is anti-symmetric, then det(A) = 0. (Hint: carefully
    use Theorem 3.5 on page 187.)


    2. Relevant equations
    Theorem 3.5: If B is obtained from A by multiplying a row (column) of A by a real number c, then det(B) = c det(A).

    A^t = inverse of A


    3. The attempt at a solution
    I found the general 3x3 antisymmetric matrix to look like this:

    [
    0 a_12 a_13
    -a_12 0 a_23
    -a_13 -a_23 0
    ]

    To find the determinant I just used the method of left and right diagonals since the matrix is 3x3, and I find it to be 0. BUT I don't know how to show this for any n x n matrix with n being odd (so it can be 5x5, 7x7, etc). I don't know where to use the theorem given in the hint either. Please help! Thanks
     
  2. jcsd
  3. Nov 11, 2008 #2

    Dick

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    Forget the specific dimension. You had better know that det(A)=det(A^t). Do you? If so, that tells you that det(A)=det(A^t)=det(-A). You can get -A from A by multiplying each row of A by -1, one at a time. Now what? Remember the number of row is odd.
     
  4. Nov 12, 2008 #3
    I see, I get to this point:

    det(-A) = (-1)^n * det(A)

    I'm not sure where to go from there though. For an odd n det(-A) will be negative of det(A), how does this show det(A) = 0? thanks
     
  5. Nov 12, 2008 #4

    gabbagabbahey

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    If I told you that [itex]x=-x[/itex], would you be able to tell me what [itex]x[/itex] was?:wink:

    P.S. You wouldn't happen to be a Utah Jazz fan would you ? :smile:
     
  6. Nov 12, 2008 #5

    Dick

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    If you've got det(A^t)=det(A) and det(A^t)=det(-A) and det(-A)=(-1)^n*det(A), and you know (-1)^n=(-1), then you have det(A)=-det(A), right? Chain them all together. For what real number is x=(-x)? There's only one.
     
  7. Nov 12, 2008 #6

    Dick

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    Utah Jazz????? I give up. I'm a sports ignoramus. Basketball player? Dwill? Am I getting close?
     
  8. Nov 12, 2008 #7

    gabbagabbahey

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    D-Will is the Nickname for Deron Williams; one of the best point guards (yes, that's basketball) in the NBA and a member of this years Gold medal winning US Olympic squad :smile:
     
  9. Nov 12, 2008 #8

    Mark44

    Staff: Mentor

    From Dick's hint you can say something about det(A^t), and from the given information, you can say something about (-1)^n.
     
  10. Nov 12, 2008 #9

    Dick

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    I feel proud I knew it was basketball. Good thing this isn't the Sports Forum. I won't ask what a "point guard" is.
     
  11. Nov 12, 2008 #10
    Haha wow I should've seen that one. :(

    And gabba yes I am a Deron Williams fan, though not a Jazz fan. :) I don't hate them or anything, just neutral. I am actually rooting for Houston, and I can't wait to see the first game between them and the Jazz this year after being eliminated by them last few years. Utah also seems to have a thing against Ron Artest, so that will be fun to watch too. :)

    Anyways, thanks a lot for the help, unfortunately I'll probably have many more questions to come since I'm really trying to catch up in my Linear Algebra class right now.
     
  12. Nov 12, 2008 #11

    Dick

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    Watch less basketball. Do more linear algebra. Wouldn't that be more fun? Just kidding.
     
  13. Nov 12, 2008 #12

    gabbagabbahey

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    I'm a Raptors fan myself, but You've got to respect a guy with Deron's talents....I was pretty big on Houston coming into the season, but after watching them get killed by the Lakers in the 4th quarter the other night, I think it's safe to say that they have an outside shot at best of getting to the finals.
     
  14. Nov 12, 2008 #13

    Dick

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    I'm outta here.
     
  15. Nov 12, 2008 #14

    gabbagabbahey

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    A healthy dose of both is my prescription :smile: (And playing basketball is even better)
     
  16. Nov 12, 2008 #15

    Dick

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    Absolutement.
     
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