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

aλ1 + b2λ + c2λ + ... + kgλn = 0,

azλ1 + bgλg + czλz + ... + kgλn = 0,
+ cλ + ... + k12 λ = 0,

anλ1 + b nλ2 + C nλg + ... + knλn = 0,

we shall obtain the following ratios between A,, Ag, λ...λ

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. 3 and 4, 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 such 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 ab ̧¤; ab ̧¤ ̧‚ 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.

2

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.

In the first place, we observe that

a1, a2 = a,b,- ab1 = − (b1a, — ba‚) = ·

b1, b2 Again,

The theorem enunciated is thus proved for determinants of two and of three rows, and may by successive inductions be extended to any number.

COR. It hence follows that, if any two lines or columns of a determinant be identical, the determinant will vanish, For we see, by the above theorem, that

Hence, if all the terms in any line or column of a determinant be multiplied by any given quantity, the determinant itself will be multiplied by the same quantity.

10. DEF. From any given determinant, other determinants may be formed, by omitting an equal number of lines and columns of the given determinants. These are termed MINORS of the given determinant, and are called first, second, &c. minors, according as one, two, &c. lines and columns have been omitted. Thus

11. To investigate the relation which must hold among the coefficients L, M, N, λ, μ, v, in order that the quadratic function

La2 + MB2+Ny2 + 2λßy + 2μya + 2vaß

may be the product of two factors of the first degree in a, B, y.

The given expression is identical with

(La + vß + μy) a + (va + MB + λy) B + (μa + λß + Ny) y.

Now, if the relation between L, M, N, λ, μ, v be such that, for all values of a, ẞ, y, the three linear functions

La + β + μη,

να + β + λγ, μα + β + Ν

may bear to one another constant ratios (p: qr, suppose), then the given expression will be the product of two factors, respectively proportional to

for all values of a, ẞ, y, and therefore for those which make the numerators of any two of the above fractions = 0. That is, values of a, B, y exist, which simultaneously satisfy the equations

La + β + γ = 0,

va + MB + λy = 0,

μα + β + γ = 0.

Hence, eliminating a, B, y, we get, as the condition that the given expression may be the product of two factors, the equation

the evanescence of which is the necessary condition that the given quadratic function may break up into two factors, is termed the Discriminant of that function.

12. PASCAL'S THEOREM.

From the analytical result stated in Art. 6 of the present chapter, that the value of a determinant is not altered by changing its lines into columns and its columns into lines, we obtain a proof of Pascal's theorem, which asserts that

If a hexagon be inscribed in a conic, and the pairs of opposite sides be produced to intersect, the points of intersection lie in the same straight line.