DETERMINANTS, an important class of algebraic functions which owe their origin to the attempt to formulate the solutions of gen eral systems of simultaneous linear equations. Snch a system of the second order is chr-Fbi,4 asx+bsy=14; from which — , aok—alei x— • aim The solution of the system of the third order, °Ix biy -1- ca=cs, I, 2, 3), in like manner gives alb.c, (h&c, ckbics with expressions of similar form for y and a. The functions appearing in the,mimerators and denominators of the expressions for the unknowns in the above, and in similar systems of equations, are determinants. They are formed in' accordance with a general principle, the first precise statement of which was based upon the recognition of the two classes of per mutations, as will presently be explained.
2. It is shown in algebra (q.v.) that the num ber of permutations of n elements arranged in a is n(te —1. . . 2. 1 is! Any two elements, whether adjacent or not, standing in their natural order in a constitute a permanence; standing in an order the, reverse of the natural, an inversion. Thus, in 'the per mutation deacb the permanences are de,, ab; the inversions, da, dc, ea, ec, eb, cb.
The permutations of any set of elements are divided into two classes, : the is which the number of inversions is even, and the negative, in which the nimiber is odd. When the elements are arranged in the natural order the ,number of inversions is zero, which is even.
3. Interchanging two adjacent elements, • and of a permutation changes its class, if a a is a permanence, a a is an inversion and vice versa; and the interchange either intro duces or destroys an inversion. When the two elements interchanged are nonadjacent let the number of elements between them be q and rep resent these, in the aggregate, by Q. As in the Preceding case the interchange has no effect upon the relatioa.of a and. a to the elements preceding or following oQa. The arrangement Qe a :lag now be obtained by interchanging a with of the q elements of Q in turn, after which a may be moved to the first place by successive interchanges with the 0-1 elements of Qa.
Hence, the total number of interchanges of ad jacent elements involved in the transition from the order aQ a to the order .aQa is 2g + 1, an odd number; from which follows the important theorem: The interchange of any two elements of a permutation changes its class.
Of any complete set of permutations half are positive and one-half negative, 4. Assume n' elements arranged in a square array thus: . . , ain9 I I . . . ch(n) I 1 • • • ..... I I (Woe ... 009 I In this array the position of any element is -shown by its indices. For examples, as"' is in the third column and the fifth row. The diagonal through ui . . . anon) is called the principal diagonal; that ihrough an-1, ... (not) the secondary diagonal; the position occupied by of the leading position.
The above array, enclosed by vertical bars as shown, is used to represent the determinant of its n' elements. This function may now be defined.
Write doWn the product of the n elements on the principal diagonal, arranging them in the natural order, thus: . . . an(n). Thii is the principal term of the determinant. Now permute the subscripts of the principal term in every possible way, leaving the superscripts un disturbed. To such of the n! resulting terms as involve the positive permutations of the sub scripts giVe the plus sign; to those involving the negative permutations, the minus sign. The algebraic sum of all the terms thus obtained is the determinant represented by the given array.
. Applying the process to the determinant array of the second order, there results ere I -.=-.alich" —ark"; a al while that of the thir'd order gives allal"a."' + atia."01"' + area"— Each term of a determinant thus contains a single element from each column and each row of its array and is, therefore, a homogeneous function of its elements.