187. Inversion. In order to find the development of determinants with more than three rows and columns, the idea of an inversion is necessary. If in a series of positive integers a greater integer precedes a less, there is said to be an inversion. Thus in the series 1 2 3 4 there is no inversion, but in the series 1 2 4 3 there is one inversion, since 4 precedes 3. In 1 4 2 3 there are two inversions, as 4 precedes both 2 and 3; while in 1 4 3 2 there are three inversions, since 4 precedes 2 and 3, and also 3 precedes 2. 188. Development of the determinant. In the development of the determinant of order three we have If we keep the order of letters in each term the same as their order in the principal diagonal (as we have done in the development above), it is observed that the subscripts in the various terms take on all possible permutations of the three digits 1, 2, and 3. The permutations that occur in the positive terms are 1 2 3, 2 31, 312, in which occur respectively 0, 2, and 2 inversions. The permutations that occur in the subscripts of the negative terms are 3 2 1, 2 1 3, 1 3 2, in which occur respectively 3, 1, and 1 inversions. Thus in the subscripts of the positive terms an even number of inversions occur, while in the subscripts of the negative terms an odd number of inversions occur. This means of determining the sign of a term of the development we shall assume in general. When we have a determinant with n rows and columns it is called a determinant of the nth order. The development of such a determinant is defined by the following RULE. The development of a determinant of the nth order is equal to the algebraic sum of the terms consisting of letters following each other in the same order in which they are found in the principal diagonal but in which the subscripts take on all possible permutations. A term has the positive or the negative sign according as there is an even or an odd number of inversions in the subscripts. This means of finding the development of a determinant is useful in practice only when the elements of the determinant are letters with subscripts such as in (2) below. When the elements are numbers we shall find the value of the determinant by a more convenient method. In this statement it is assumed that the number of inversions in the subscripts of the principal diagonal is zero. If this number of inversions is not zero, the sign of any term is + or according as the number of inversions in its subscripts differs from the number in the subscripts of the principal diagonal by an even or an odd number. Since each term contains every letter a, b, k and also every index 1, 2, ・, n, one element of each row and column occurs in each term. the terms a2b4c1d3 and a4b2c3d1, for instance, have the minus sign, as 2 4 1 3 has three inversions and 4 2 3 1 has five inversions; while the terms a1b4c2d3 and abg cod1 have two and six inversions respectively and hence have the positive sign. 189. Number of terms. We apply the theorem of permutations to prove the following THEOREM. A determinant of the nth order has n! terms in its development. Since the number of terms is the same as the number of permutations of the n indices taken all at a time, the theorem follows immediately from the corollary on p. 145. 190. Development by minors. In the development of the determinant of order three, p. 208, we may combine the terms as follows: is We observe that the coefficient of a, is the determinant that we obtain by erasing the row and column in which a, lies. A similar fact holds for the coefficients of a2 and ag. The determinant obtained by erasing the row and column in which a given b1 c1 element lies is called the minor of that element. Thus ხვ C3 the minor of a2. We notice that in the above development by minors (1) the sign of a given term is or according as the + sum of the number of the row and the number of the column of the element in that term is even or odd. Thus in the first term a, is in the first row and the first column, and since 1 + 1 = 2, the statement just made is verified for that case. Similarly, a, is in the first column and the second row, and since 1+2 = 3 is odd, the sign is minus and the law holds here. The last term is positive, which we should expect since a is in the first column and the third row, and 1 + 3 = 4. The proof for the general validity of this law of signs is found on p. 215. The elements of any other row or column than the first may be taken and the development given in terms of the minors with respect to such elements. For instance, take the development with respect to the elements of the second row, The rule of signs is the same as given above; that is, for instance, the last term is negative, as c2 is in the third column. and the second row, and 2 + 3 = 5. By generalizing these considerations we may find the development of a determinant by minors by the following RULE. Write in succession the elements of any row or column, each multiplied by its minor. Give each term a + or a sign according as the sum of the number of the row and the number of the column of the element in that term is even or odd. Develop the determinant in each term by a similar process until the value of the development can be determined directly by multiplication. That this rule for development gives the same result as the definition given in § 188 we have seen for a determinant of order three. The fact holds in general, as we shall prove (p. 215). EXERCISES 1. In the determinant of order four on p. 209 what sign should be prefixed to the following terms? (a) c4b3a2dı. Solution: c4b3a2d1 = a2b3c4d1. In 2 3 4 1 there are three inversions. The sign should be minus. 2. Develop by minors the following and find the value of the determinant. = 3.(6 — 4) — 2 (12 − 4) + 3 (8 − 4) = 6 − 16 + 12 = 2. |