Comparing the values of x and y it is observed that: 1. They have the same denominator. 2. The numerator of the value of x may be formed from the denominator by replacing the coefficients of x by the corresponding known terms k1 and k 3. The numerator of the value of y may be formed from the denominator by replacing the coefficients of y by the corresponding known terms k1 and k The common denominator a1b — ab1 is called the determinant of the system. A convenient symbol for a,b,- ab1, suggested by the arrangement in (1) of the coefficients of x and y in two columns and two rows, is called a determinant of the second order. ab - ab1 is called the developed form, or the development, of this determinant. a1, ɑ2, b1, b2 are called its elements. NOTE. - Some authors employ the terms element and constituent with the meanings here given to constituent and element, respectively. 544. To develop a determinant of the second order. The second member may be written bα1 - b1a2, or 1. The positive term, a,b, or ba1, is obtained by multiplying the element a, in the first column and first row by the element b in the next column and next row; or by multiplying the element b, in the second column and second row by the element a, in the preceding column and preceding row. The selection of an element from any column or row before the selection of an element from a preceding column or row constitutes an inversion. Then, the positive term formed in the first way presents no inversions, but formed in the second way presents two inversions, namely, the selection of an element from the second column before that of an element from the preceding column, and the selection of an element from the second row before that of an element from the preceding row. In either case the positive term of the development presents an even number of inversions. 2. The negative term, abi, or ba, is obtained by multiplying the element a, in the first column and second row by the element b in the second column and first row, and making the product negative; or by selecting the elements in the reverse order and making the product negative. In the first way there is an inversion of rows, in the second way, an inversion of columns. In either case the negative term of the development presents an odd number of inversions. 545. Any square array of n elements arranged in n columns and n rows represents a determinant of the nth order. In harmony with the principles of the preceding article a determinant of any order is now defined as a square array of numbers that, by common agreement, represents the algebraic sum of all the products, or constituents, that can be formed by taking one element, but not more than one, from each column and from each row, making constituents that present an even number of inversions positive and constituents that present an odd number of inversions negative. 546. Development of any determinant. Let a b2 Co be a determinant of the third order. аз вз C3 By the definition of a determinant, each constituent of this determinant contains three elements as factors, one and only one taken from each column and from each row. - Hence, the constituents involving a, are a,b,c, and a,b,c, the latter being negative because it presents one inversion. fore, the sum of the constituents involving a1 is There which may be obtained from the given determinant by canceling or deleting the elements that cannot be associated with a b3 C3 multiplied is called the minor of the element a. When the minor is given the proper sign, in this case +, it is called the co-factor of the element. Similarly, the sum of the constituents involving a2 is derived by deleting the elements that cannot be associated with a2, Similarly, the sum of the constituents involving a, is Since each constituent of the given determinant must involve either a, a, or ag, we have found all the constituents. Hence, The same result is obtained by using any column or any row of elements as the first column is used above. For example, selecting the elements of the second column, which is the former result differently arranged. The above discussion applies to a determinant of any order. Hence, The development of a determinant of any order is equal to the algebraic sum of the products of the elements of any column or row and their respective co-factors. 547. The minors corresponding to the elements a1, α, b1, by,, are denoted by A1, A2 ..., B1, B2, 548. Number of constituents. .... Since the co-factors of each of the n elements in any selected column or row of a determinant of the nth order are determinants of the (n - 1)th order, a determinant of the nth order has n times as many constituents as a determinant of the (n − 1)th order; this, in turn, has (n − 1) times as many constituents as a determinant of the (n − 2)th order; and so on, until a determinant of the 2d order is reached, which has 2 constituents. Hence, - A determinant of the nth order has n(n − 1)(n − 2) ..... 2, or [n, -- constituents. SOLUTION. Multiplying the elements of the first column by their co-factors, and adding, the given determinant is reduced to Proceeding as in Ex. 1, the given determinant is reduced to 6 9 8 2 3 4 2 3 4 1 10 11 12 5 10 11 12 +9 6 9 8-3 6 9 8 14 15 16 14 15 16 2 3 4 Since by Ex. 1 the first determinant is equal to 16, the given determinant * For economy of space the sign of a negative element may be written above the element. |