From the last three of these, we have as in the preceding article the expression on the left being a determinant of the fourth order. Also we see that the coefficients of a,, b, c, d, taken with their proper signs are the minors obtained by omitting the row and column which respectively contain these constituents. 501. More generally, if we have n homogeneous linear equations involving n unknown quantities x,, x, x,... x, these quantities can be eliminated and the result expressed in the form The left-hand member of this equation is a determinant which consists of n rows and n columns, and is called a determinant of the nth order. The discussion of this more general form of determinant is beyond the scope of the present work; it will be sufficient here to remark that the properties which have been established in the case of determinants of the second and third orders are quite general, and are capable of being extended to determinants of any order. For example, the above determinant of the nth order is equal to according as we develop it from the first row or the first column. Here the capital letters stand for the minors of the constituents denoted by the corresponding small letters, and are themselves determinants of the (n-1)th order. Each of these may be expressed as the sum of a number of determinants of the (n-2)th order; and so on; and thus the expanded form of the determinant may be obtained. Although we may always develop a determinant by means of the process described above, it is not always the simplest method, especially when our object is not so much to find the value of the whole determinant, as to find the signs of its several elements. 502. The expanded form of the determinant and it appears that each element is the product of three factors, one taken from each row, and one from each column; also the signs of half the terms are + and of the other half. The signs of the several elements may be obtained as follows. The first element a,b,c,, in which the suffixes follow the arithmetical order, is positive; we shall call this the leading element; every other element may be obtained from it by suitably interchanging the suffixes. The sign + or is to be prefixed to any element ac cording as it can be deduced from the leading element by an even or odd number of permutations of two suffixes; for instance, the element a,b,c, is obtained by interchanging the suffixes 1 and 3, therefore its sign is negative; the element a,b,c, is obtained by first interchanging the suffixes 1 and 3, and then the suffixes 1 and 2, hence its sign is positive. 503. The determinant whose leading element is a,b,c,d, ... may thus be expressed by the notation theplaced before the leading element indicating the aggregate of all the elements which can be obtained from it by suitable interchanges of suffixes and adjustment of signs. Sometimes the determinant is still more simply expressed by enclosing the leading element within brackets; thus (a,b,c,d, ...) is used as an abbreviation of ± a ̧b2c ̧d ̧.... Example. In the determinant (a,b,c,d) what sign is to be prefixed to the element abзc1de? From the leading element by permuting the suffixes of a and d we get abcde; from this by permuting the suffixes of b and c we have a ̧ ̧¤‚ ̧е; by permuting the suffixes of c and d we have a,b,c,de; finally by permuting the suffixes of d and e we obtain the required element abcde; and since we have made four permutations the sign of the element is positive. 504. If in Art. 501, each of the constituents b, c,,... k1 is equal to zero the determinant reduces to a,4,; in other words it is equal to the product of a, and a determinant of the (n - 1)th order, and we easily infer the following general theorem. If each of the constituents of the first row or column of a determinant is zero except the first, and if this constituent is equal to m, the determinant is equal to m times that determinant of lower order which is obtained by omitting the first column and first row. Also since by suitable interchange of rows and columns any constituent can be brought into the first place, it follows that if any row or column has all its constituents except one equal to zero, the determinant can immediately be expressed as a determinant of lower order. This is sometimes useful in the reduction and simplification of determinants. Diminish each constituent of the first column by twice the corresponding constituent in the second column, and each constituent of the fourth column by three times the corresponding constituent in the second column, and we obtain and since the second row has three zero constituents this determinant 505. The following examples shew artifices which are occasionally useful. Example 1. Prove that d = (a+b+c+d) (a − b + c − d) (a − b − c + d) (a+b− c –d). d с d a b d с b a By adding together all the rows we see that a+b+c+d is a factor of the determinant; by adding together the first and third rows and subtracting from the result the sum of the second and fourth rows we see that a-b+c-d is also a factor; similarly it can be shewn that a-b-c+d and a+b-c-d are factors; the remaining factor is numerical, and, from a comparison of the terms involving a on each side, is easily seen to be unity; hence we have the required result. Example 2. Prove that 1 1 1 1 = (ab) (ac) (a–d) (b −c) (b − d) (c–d). a b с d a2 b2 c2 d2 a3 b3 c3 d3 The given determinant vanishes when b=a, for then the first and second columns are identical; hence a - b is a factor of the determinant [Art. 514]. Similarly each of the expressions ac, a- -d, b-c, b-d, cd is a factor of the determinant; the determinant being of six dimensions, the remaining factor must be numerical; and, from a comparison of the terms involving bc2d3 on each side, it is easily seen to be unity; hence we obtain the required result. |