Page 21 - Vector Analysis

§1.5 Determinants 17

thus (3, 1, 2) is an even permutation of (1, 2, 3). On the other hand, (1, 3, 2) is obtained by
interchanging pairs of (1, 2, 3) once:

(1, 2, 3) ÝÑτ(2,3) (1, 3, 2);

thus (1, 3, 2) is an odd permutation of (1, 2, 3).

11

23 23

Even permutations Odd permutations

Figure 1.1: Even and odd permutations of degree 3

For n = 3, the even and odd permutations can also be viewed as the orientation of the
permutation (k1, k2, k3). To be more precise, if (1, 2, 3) is arranged in a counter-clockwise
orientation (see Figure 1.1), then an even permutation of degree 3 is a permutation in the
counter-clockwise orientation, while an odd permutation of degree 3 is a permutation in the
clockwise orientation. From figure 1.1, it is easy to see that (3, 1, 2) is an even permutation
of degree 3 and (1, 3, 2) is an odd permutation of degree 3.

Definition 1.59 (The permutation symbol). The permutation symbol εk1k2¨¨¨kn is a function
of permutations of degree n defined by

"1 if (k1, k2, ¨ ¨ ¨ , kn) is an even permutation of degree n,
εk1k2¨¨¨kn = ´1 if (k1, k2, ¨ ¨ ¨ , kn) is an odd permutation of degree n.

Remark 1.60. One can extend the domain the permutation symbol to all the sequence
(k1, k2, ¨ ¨ ¨ , kn) by defining that εk1k2¨¨¨kn = 0 if (k1, k2, ¨ ¨ ¨ , kn) is not a permutation of degree
n.

Definition 1.61 (Determinants). Given an n ˆ n matrix A = [aij], the determinants of A,
denoted by det(A), is defined by

n

det(A) = ÿ ź

εk1k2¨¨¨kn aℓkℓ .

(k1,¨¨¨ ,kn)PP(n) ℓ=1

n

We note that the product ś aℓkℓ in the definition of the determinant is formed by

ℓ=1

multiplying n-elements which appears exactly once in each row and column.