It will be seen that the above process is really equivalent to that of eliminating λ,, λ, λ, between the equations (A) and (B). Hence

To effect the elimination, multiply the equations in order by A, A, A, A, add them, and equate the coefficients of x2, x, x severally to zero. We shall then have

а1λ ̧ + b1λ1⁄2 + e̟ ̧λ ̧ + d ̧λ ̧ = 0 ........................ (B').

To determine the three ratios λ : μ2 : λ : λ, multiply equations (A') in order by μ, Mg, μ, add, and equate to zero the coefficients of A, A. We thus get

Now, treating equations (C') as equations (A) were treated, we see that

These equations may be more conveniently written in the following equivalent forms:

Eliminating by means of these equations λ, λ, λg, λ4, from equation B', we get, as the result of the elimination of x1, x2, x, x between the four given equations,

And since the above process is equivalent to the elimination of 1⁄4, 1⁄2, λg, 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 horizontal and vertical rows, it may be similarly proved that

It may also be proved that, if we have n-1 equations connecting n quantities A,, A...λ, such as

we shall obtain the following ratios between A, λg, λ ̧‚.........λ‚‚

For,

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. 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." 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 first by an odd number of permutations has a negative sign. Every term formed by an even number of permutations has a positive sign."

Thus, it will be observed that the terms a,b,c2, ɑ ̧ ̧¤ ̧‚ 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 a,b,c,, 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 abcd ̧, derived by three, has a negative sign.

8. The sign of a determinant is changed by interchanging any two consecutive lines or columns.