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A Determinant's relation to permutations

  1. Apr 27, 2010 #1
    for a 3 x 3 matric of values

    a11 a12 a13
    b21 b22 b23
    c31 c32 c33

    the determinant will be a11a22a33+a12a23a31+a13a21a32-a13a22a31-a12a21a33-a11a23a32

    the last three are negative because they are odd permutations. The first three are even permutations

    A permutation apparently is found by the number that are out of order, order being 1,2,3,4 increasing.

    4,2,1,3 would result in a 2+1=3 permutation I believe.

    How does all of this fit together? I do not understand why permutations matter in relation to the determinant, which fits into the inverse and so on.
  2. jcsd
  3. Apr 27, 2010 #2


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    What you have there is a special case of the more general statement:

    [tex]\det A=\sum_P \sigma(P) A_{1,P1}\cdots A_{n,Pn}[/tex]

    where the sum is over all permutations P on the set {1,2,...,n} and σ(P) is =1 if the permutation is even, and =-1 if the permutation is odd. (I usually write (-1)P instead of σ(P), but that notation confuses most people).

    I don't know if there's a short answer to the question of how it all fits together. Can you ask a more specific question?
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