Section 2.4

A Combinatorial Approach to Determinants

PERMUTATIONS

A permutation of the integers {1, 2, 3, . . . , n} is an arrangement of these integers in some order without omissions or repetitions.

Example: (2, 5, 1, 3, 4) is a permutation of the integers {1, 2, 3, 4, 5}.

Notation: We denote a general permutation by( j1, j2, . . . , jn).

INVERSIONS

An inversion is said to occur in a permutation ( j1, j2, . . . , jn) whenever a larger integer precedes a smaller one.

FINDING THE TOTAL NUMBER OF INVERSIONS IN A PERMUTATION

1. Find the number of integers that are less than j1 and that follow j1 in the permutation.

2. Find the number of integers that are less than j2 and that follow j2 in the permutation.

3. Continue the process.

EVEN AND ODD PERMUTATIONS

• A permutation is called even if total number of inversions is an even integer.

• A permutation is called odd if the total number of inversion is an odd integer.

ELEMENTARY PRODUCTS

By an elementary product from an n×n matrix A we shall mean any product of n entries, no two of which come from the same row or same column.

Notation: We will denote a general elementary product by

njnjj aaa 21 21

SIGNED ELEMENTARY PRODUCTS

A signed elementary product from A is an elementary product that is

multiplied by +1 if the permutation ( j1, j2, . . . , jn) is even and by −1 if the permutation is odd.

njnjj aaa 21 21

EXAMPLE

Elementary Product

Associated Permutation

Even or Odd Signed Elem. Prod.

a11a22a33 (1, 2, 3) even a11a22a33

a11a23a32 (1, 3, 2) odd −a11a23a32

a12a21a33 (2, 1, 3) odd −a12a21a33

a12a23a31 (2, 3, 1) even a12a23a31

a13a21a32 (3, 1, 2) even a13a21a32

a13a22a31 (3, 2, 1) odd −a13a22a31

THE DEFINITION OF DETERMINANT IN TERMS OF ELEMENTARY PRODUCTS

Let A be a square matrix. We define det(A) to be the sum of all signed elementary products from A.

SHORTCUT FOR COMPUTING 2×2 AND 3×3 DETERMINANTS

The determinants can be computed by adding the products on the “forward” diagonals and subtracting the products on the “backward” diagonals.

SYMBOLIC NOTATION FOR THE DETERMINANT

The determinant of A is may be written symbolically as

where Σ indicates that the terms are to be summed over all permutations ( j1, j2, . . . , jn) and the + or − is selected in each term according the whether the permutation is even or odd.

njnjj aaaA

21 21)det(