A Determinant's relation to permutations

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SUMMARY

The discussion centers on the relationship between determinants and permutations in a 3x3 matrix. The determinant is calculated using the formula: det A = a11a22a33 + a12a23a31 + a13a21a32 - a13a22a31 - a12a21a33 - a11a23a32, where the signs of the terms depend on whether the corresponding permutations are even or odd. The concept of permutations is crucial as it determines the sign of each term in the determinant calculation, with even permutations contributing positively and odd permutations negatively. The general formula for determinants is expressed as det A = ∑_P σ(P) A_{1,P1}...A_{n,Pn}, where σ(P) indicates the parity of the permutation.

PREREQUISITES
  • Understanding of matrix algebra, specifically 3x3 matrices.
  • Familiarity with the concept of permutations and their properties.
  • Knowledge of determinant calculations and their significance in linear algebra.
  • Basic understanding of even and odd permutations.
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  • Study the properties of determinants in linear algebra.
  • Learn about the role of permutations in combinatorial mathematics.
  • Explore the applications of determinants in solving linear equations.
  • Investigate advanced topics such as multilinear algebra and its relation to determinants.
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Students and professionals in mathematics, particularly those studying linear algebra, matrix theory, and combinatorial mathematics. This discussion is beneficial for anyone seeking to understand the foundational concepts of determinants and their applications.

vjk2
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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.
 
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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?
 

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