These equations may be more conveniently written in the following equivalent forms: 29 Eliminating by means of these equations A, A, λ 3, λ 42 from equation B', we get, as the result of the elimination of x1, x, x, x between the four given equations, And since the above process is equivalent to the elimination of ,,,, λ between the equations (A') and (B′), we see that 6. The law of formation will be sufficiently obvious from the above investigations. If we have n lines and columns, it may be similarly proved that DETERMINANTS OF FOUR ROWS. 65 It may also be proved that if we have n-1 equations connecting n quantities A, A... An, such as ɑ1⁄2 λ + b2λ1⁄2 + c2λ, + + k2λn=0, ... : an λ1 + b 1λg + C nλg + ... + knλn = 0, we shall obtain the following ratios between A, λq, λg... Nn it will be seen that the former contains 1. 2 or two terms, the latter 1.2.3 or six. It may also be proved that, if n quantities be eliminated from n linear homogeneous equations, the resulting determinant will contain 1.2.3...n terms. For, referring to the relation between determinants of n and n-1 rows, given in Arts. (4), (5), (6), it will be seen that this theorem is true for a determinant of n rows, if it be true for one of n 1. But it is true for three rows, therefore it is universally true. 7. The horizontal rows of a determinant are commonly spoken of as "lines," the vertical ones as "columns." It will be observed, moreover, that each term is the product of n factors, one taken from each line and from each column, and that the coefficients of one half of the terms are +1, of the other 1. To determine the sign of any particular term we proceed as follows. Considering for simplicity the case of three rows, we have Here we observe, first, that (the factors of each term being arranged in alphabetical order, that is, in the order of the columns) the term a,b,c, (in which the suffixes follow the arithmetical order, that is, the order of the lines) has a positive coefficient. Now every other term may be formed from this by making each suffix change places with either of its adjacent suffixes a sufficient number of times. Thus the term a,b,c, is produced by simply making the suffixes 2 and 3 exchange places. The term a,b,c, is produced by making the suffix 3 change places, first with 2, and next with 1, which is then adjacent to it. If this process of interchanging the suffixes of two consecutive letters be called a "permutation," we may enunciate the following law, which by inspection will be seen to hold. "Every term derived from the permutations has a negative sign. even number of permutations has a positive sign.' first by an odd number of Every term formed by an Thus, it will be observed that the terms a,b,c,; a,b,c, each of which is derived from a,b,c, by one permutation, have negative signs. The terms a,b,c,; a,b,c,, each formed by two permutations, have positive signs. The term abe̟, formed by three permutations, has a negative sign. In like manner, in the case of a determinant of four rows, if a,b,c,d, have a positive sign, such a term as a,b,c,d, derived by two permutations, will have a positive sign, while a,b,c,d,, derived by three, has a negative sign. 8. The sign of a determinant is changed by interchanging any two consecutive lines or columns. |