Page 20 - Vector Analysis

16 CHAPTER 1. Linear Algebra

1.5 Determinants

In order to introduce the notion of the determinant of square matrices, we need to talk
about permutations first. Note that there are many other ways of defining determinants,
but it is quite elegant to use the notion of permutations, and we can derive a lot of useful
results via this definition.

Definition 1.56 (Permutations). A sequence (k1, k2, ¨ ¨ ¨ , kn) of positive integers not ex-
ceeding n, with the property that no two of the ki are equal, is called a permutation of
degree n. The collection of all permutations of degree n is denoted by P(n).

A sequence (k1, k2, ¨ ¨ ¨ , kn) can be obtained from the sequence (1, 2, ¨ ¨ ¨ , n) by a finite
number of interchanges of pairs of elements. For example, if k1 ‰ 1, we can transpose 1
and k1, obtaining (k1, ¨ ¨ ¨ , 1, ¨ ¨ ¨ ). Proceeding in this way we shall arrive at the sequence
(k1, k2, ¨ ¨ ¨ , kn) after n or less such interchanges of pairs.

In general, a permutation (k1, k2, ¨ ¨ ¨ , kn) can be expressed as

τ(iN ,jN ) ¨ ¨ ¨ τ(i2,j2)τ(i1,j1)(1, 2, ¨ ¨ ¨ , n) = (k1, k2, ¨ ¨ ¨ , kn),

where τ(i,j) is a “pair-interchange operator” which swaps the i-th and the j-th elements (of
the object fed into), and N is the number of pair interchanges. We call such pair-interchange
operators the permutation operator. Since τ(i,j) is the inverse operator of itself, we also have

τ τ(i1,j1) (i2,j2) ¨ ¨ ¨ τ(iN ,jN )(k1, k2, ¨ ¨ ¨ , kn) = (1, 2, ¨ ¨ ¨ , n).

We remark here that the number of pair interchanges (from (1, 2, ¨ ¨ ¨ , n) to (k1, k2, ¨ ¨ ¨ , kn))
is not unique; nevertheless, if two processes of pair interchanges lead to the same permuta-
tion, then the numbers of interchanges differ by an even number. This leads to the following

Definition 1.57 (Even and odd permutations). A permutation (k1, ¨ ¨ ¨ , kn) is called an
even (odd) permutation of degree n if the number required to interchange pairs of
(1, 2, ¨ ¨ ¨ , n) in order to obtain (k1, k2, ¨ ¨ ¨ , kn) is even (odd).

Example 1.58. If n = 3, the permutation (3, 1, 2) can be obtained by interchanging pairs

of (1, 2, 3) twice:

(1, 2, 3) ÝÑτ(1,3) (3, 2, 1) ÝÑτ(2,3) (3, 1, 2);