DETERMINANTS (I.at. dcterminans, pres. part. of di ter-minor.% front dr- + liminarc, to bound. limit 1. Certain algebraic functions re markable for their brevity of notation and their wealth of significant properties. h0.. • a,bt, or are nerdy other notations for a b„ The first form is commonly used in ment a ry mat hematics. jut ila rly a, It, c,1 a., b., a, b, represents a,kr + (1,1, r, -4- a — a It e Sn•il funetiens an. called deter minants. and the quantities a„ e,. a„„ . . .
are ealled elvni•nt The first or square form of notation is called the array notation. if there arc more column- than rows the form is ealled matrir. An array of two eohnnns and two rows is called a determinant of the second order, one of three eoluitins and three 1•atts a determinant of the third order, on. In the expansion of a determinant of the second order there arc two terms, each containing the letters a, it, but differing in the arrangement of the sub scripts. I. 2. In the expansion of a determinant of the third order there are six terms, each con taining the letters e, 1,, e, but differing in the arrangement of the subscript- 1. •, 3. Thus there are two terms in the expansion of a deter minant of the second order, and six in one of the third order, half of which are positive and half negative. The signs are selected accenting to the arrangement of the subscript,. If there is an even number of inversions in the order of the subscripts of any term, its sign is considered plus; if an odd number, its sign is considered minus: e.g. in the term a 3 standing before 2 is an inversion, since the natural order is 2 before 3; likewise, 3 before I and 2 before 1 are inversions; there being three inversion, the sign is A determinant of the fourth order contains 16 elements. and its expansion contains 24 terms. In general. a determinant of the nth order contains rt= elements, and it- expan sien n (a — 1) 2) .. . . 2 • I. or n! terms. Some of the leading of detc•tninant-z are: (1) If two adja•ent columns or rows of a determinant are interchanged, the sign only of the determinant is changed; if a column or row is transposed over an odd number of columns or rows, the sign only of the determinant is changed; if a column et- row is transposed over an even number of columns or rows, the determi• pant is unchanged. (2) If the columns be made
rows, and conversely, the determinant is un changed. (:1) If the elements of a row or column be added to the corresponding elements of an other column or row, the value of the determi nant is nnehanged. (.1) Multiplying the ele ments of any column or IOW by a 1111110A '1' multi plies the determinant by that number. (5) If two columns or two rows are identical, or if the elements of any row or column arc all zeros, the determinant vanishes. Snell properties are of great a id in evaluating del ermina nt s.
Various methods of expansion have been de by which the terms, each containing one element and only one, from each colunm and each row. can readily and systematically be formed. While these methods can best be ob tained from a text-book on the subject, minor determinants will lie explained here, since they furnish a simple means for expanding determi mints of degrees higher than the third. If the column and row to which any clement belongs are suppressed. the resulting determinant is (-ailed a first minor of the given determinant, or the eo factor of the given clement. If two rows and two colimins be suppre.ssed. the resulting determinant is called a seennd minor of the given determinant. and so on. Thus, in c, 1.1 r, d.
A .
a, c, (7, a, 7,, c, d, (kr.11,1 is a first minor of A and a co-factor of the lending element a„ and (a,b,) is a :wenn(' miner If the co-factors of a„ b„ r„ ftl, he denoted IT A„ Il,. C,, D„ then it can he shown that A t/..D„ it should be observed that Ii .. . . involve their own sights, and that in the above equation and D, are negative. Thus a determinant of the ath order may he expanded in terms of determinants of the (a — and so on. Determi nants also admit. in their abridged form of no tation, of the fundamental operations of addi tion, subtrawtion, and multiplication (including involution ) .