1. Section 2.4 A Combinatorial Approach to Determinants

2. 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 ( j 1, j 2,..., j n ).

3. INVERSIONS An inversion is said to occur in a permutation ( j 1, j 2,..., j n ) whenever a larger integer precedes a smaller one.

4. FINDING THE TOTAL NUMBER OF INVERSIONS IN A PERMUTATION 1.Find the number of integers that are less than j 1 and that follow j 1 in the permutation. 2.Find the number of integers that are less than j 2 and that follow j 2 in the permutation. 3.Continue the process.

5. 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.

6. 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

7. SIGNED ELEMENTARY PRODUCTS A signed elementary product from A is an elementary product that is multiplied by +1 if the permutation ( j 1, j 2,..., j n ) is even and by −1 if the permutation is odd.

8. EXAMPLE Elementary Product Associated Permutation Even or OddSigned Elem. Prod. a 11 a 22 a 33 (1, 2, 3)evena 11 a 22 a 33 a 11 a 23 a 32 (1, 3, 2)odd−a 11 a 23 a 32 a 12 a 21 a 33 (2, 1, 3)odd−a 12 a 21 a 33 a 12 a 23 a 31 (2, 3, 1)evena 12 a 23 a 31 a 13 a 21 a 32 (3, 1, 2)evena 13 a 21 a 32 a 13 a 22 a 31 (3, 2, 1)odd−a 13 a 22 a 31

9. 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.

10. 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.

11. 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 ( j 1, j 2,..., j n ) and the + or − is selected in each term according the whether the permutation is even or odd.