To fix the signs in terms in the expansion of a determinant of any order, the notion of an inversion is introduced. If, in an arrangement of positive integers, a greater precedes a less, there is said to be an inversion. Thus in the order 12543, there are three inversions: 5 before 4, 5 before 3, 4 before 3. In 2341576, there are four inversions. When applied to any term in the expansion of a determinant such as (7), we say there is an inversion of the order of the subscripts presents an inversion when the letters (apart from the subscripts) have the order abcd…1 of the principal diagonal. With respect to determinants of orders 2 and 3, it may be observed that the number of inversions is even when the term is positive, and that the number of inversions is odd when the term is negative.
Consistently with these conditions, we lay down the following
Definition. A square array of n² elements, such as has been considered in the cases n=2 and n=3, is called a determinant of the nth order. It is an abbreviation for the algebraic form of all the different products that can be formed by taking as factors one and only one element from each column and each row of the array, and giving to each term a positive or negative sign according as the number of inversions of the subscripts of the term is even or odd, when the letters have the same order as the principal diagonal.
Excerpted from Introductory College Alegbra by H.L. Rietz and A.R. Crathorne. Published by Henry Holt and Company, New York, 1923.