# A Determinant's relation to permutations

1. Apr 27, 2010

### vjk2

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. Apr 27, 2010

### Fredrik

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

$$\det A=\sum_P \sigma(P) A_{1,P1}\cdots A_{n,Pn}$$

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?