2 4 Permutation n n A permutation of

2 -4 Permutation n n A permutation of the set of integers {1, 2, …, n} is an arrangement of these integers in some order without omission repetition Example 1 q n There are six different permutations of the set of integers {1, 2, 3}: (1, 2, 3), (2, 1, 3), (3, 1, 2), (1, 3, 2), (2, 3, 1), (3, 2, 1). Example 2 q List all permutations of the set of integers {1, 2, 3, 4} 2 1 3 2 1 4 3 3 4 1 4 2 1 4 3 2 3 4 2 3 3 4 1 3 2 4 1 2 2 3 1 2 4 3 4 2 3 2 4 3 4 1 3 1 4 2 4 1 2 1 3 2 3 1 2021/10/29 Elementary Linear Algebra 1

2 -4 Inversion n An inversion is said to occur in a permutation (j 1, j 2, …, jn) whenever a larger integer precedes a smaller one. n The total number of inversions occurring in a permutation can be obtained as follows: n n n 2021/10/29 Find the number of integers that are less than j 1 and that follow j 1 in the permutation; Find the number of integers that are less than j 2 and that follow j 2 in the permutation; Continue the process for j 1, j 2, …, jn. The sum of these number will be the total number of inversions in the permutation Elementary Linear Algebra 2

2 -4 Example 3 n Determine the number of inversions in the following permutations: q q q n (6, 1, 3, 4, 5, 2) (2, 4, 1, 3) (1, 2, 3, 4) Solution: q q q The number of inversions is 5 + 0 + 1 + 1 = 8 The number of inversions is 1 + 2 + 0 = 3 There no inversions in this permutation 2021/10/29 Elementary Linear Algebra 3

2 -4 Classifying Permutations n A permutation is called even if the total number of inversions is an even integer and is called odd if the total inversions is an odd integer n Example 4 q The following table classifies the various permutations of {1, 2, 3} as even or odd Permutation Number of classification Inversions (1, 2, 3) (1, 3, 2) (2, 1, 3) (2, 3, 1) (3, 1, 2) (3, 2, 1) 2021/10/29 Elementary Linear Algebra 0 1 1 2 2 3 even odd 4

2 -4 Elementary Product n n By an elementary product from an n n matrix A we shall mean any product of n entries from A, no two of which come from the same row or same column. Example q The elementary product of the matrix is 2021/10/29 Elementary Linear Algebra 5

2 -4 Signed Elementary Product n An n n matrix A has n! elementary products. There are the products of the form a 1 j 1 a 2 j 2 ··· anjn, where (j 1, j 2, …, jn) is a permutation of the set {1, 2, …, n}. n By a signed elementary product from A we shall mean an elementary a 1 j 1 a 2 j 2 ··· anjn multiplied by +1 or -1. q We use + if (j 1, j 2, …, jn) is an even permutation and – if (j 1, j 2, …, jn) is an odd permutation 2021/10/29 Elementary Linear Algebra 6

2 -4 Example 6 n List all signed elementary products from the matrices 2021/10/29 Elementary Product Associated Permutation Even or Odd Signed Elementary Product a 11 a 22 a 12 a 21 (1, 2) (2, 1) even odd a 11 a 22 - a 12 a 21 Elementary Product Associated Permutation Even or Odd Signed Elementary Product a 11 a 22 a 33 a 11 a 23 a 32 a 12 a 21 a 33 a 12 a 23 a 31 a 13 a 21 a 32 a 13 a 22 a 31 (1, 2, 3) (1, 3, 2) (2, 1, 3) (2, 3, 1) (3, 1, 2) (3, 2, 1) even odd a 11 a 22 a 33 - a 11 a 23 a 32 - a 12 a 21 a 33 a 12 a 23 a 31 a 13 a 21 a 32 - a 13 a 22 a 31 Elementary Linear Algebra 7

2 -4 Determinant n n Let A be a square matrix. The determinant function is denoted by det, and we define det(A) to be the sum of all signed elementary products from A. The number det(A) is called the determinant of A Example 7 2021/10/29 Elementary Linear Algebra 8

2 -4 Using mnemonic for Determinant n The determinant is computed by summing the products on the rightward arrows and subtracting the products on the leftward arrows n Remark: q This method will not work for determinant of 4 4 matrices or higher! 2021/10/29 Elementary Linear Algebra 9

2 -4 Example 8 n Evaluate the determinants of 2021/10/29 Elementary Linear Algebra 10

2 -4 Notation and Terminology n n n We note that the symbol |A| is an alternative notation for det(A) The determinant of A is often written symbolically as det(A) = a 1 j 1 a 2 j 2 ··· anjn where indicates that the terms are to be summed over all permutations (j 1, j 2, …, jn) and the + or – is selected in each term according to where the permutation is even or odd Remark: q 4 4 matrices need 4! = 24 signed elementary products q 10 10 determinant need 10! = 3628800 signed elementary products! q Other methods are required. 2021/10/29 Elementary Linear Algebra 11