745. Fourth Property.-A determinant of any order = 0 if it has two columns or two rows that are exactly alike.

First Proof: One can verify the theorem for any case.

Second Proof: Let the determinant with two columns alike be

By interchanging the first and second columns we get (§744):

Similarly it can be proved that a determinant of the fourth or higher orders is zero if it has two rows that are exactly alike.

This theorem may be stated as follows: If the elements of a column or a row of a determinant of any order are replaced respectively by the elements of a parallel column or row the determinant =0. For example, if the first row (1, x, y) in the determinant

is replaced by the elements of the second row (1, x, y) or those of the third row (1, x, y), the determinant

=

0 (8745).

COROLLARY. A determinant of any order is equal to 0, if the elements of a column or row have the same common factor whose removal leaves a column or row the same as one of the remaining parallel columns or rows. For, on removing the factors from the elements of the given columns or rows we multiply the resulting determinant respectively by these factors, but the resulting determinant has two columns or rows alike and is therefore zero (1745).

Without expanding the determinant prove that a and b are roots of the quadratic equation in written in determinant form,

746. Definition. -The coefficient of an element of a determinant, or more precisely, the entire multiplier of an element in a determinant, is called the minor determinant of this element of the given determinant. The minor determinants of

(3) or

A

▲ = c(abab) + c2 (b1a−ba1) + ¢ ̧ (a,b,— ab1),

with respect to the nine elements

1 2

B2, B2, B2; C3, C3, C3.

27

2

(2), (3) that

B2

=

=

b.

b3

a

3

a1

b

1

3

RULE.—To find the minor determinant of any element strike out the column and row having the element in common. The minors of the elements in the two diagonals (A, B, Câ; A, B, C) of the given determinants are plus, the signs of the other minors are minus.

COROLLARY.-The minor of any element is independent of the elements of the column and row to which this element belongs, and therefore remains unchanged if the elements of the column and row to which the element belongs are replaced by other numbers.

747. Properties of the Minor Determinants of A.-These are expressed in the following 18 equations:

The truth of these equations follows readily from direct calculation. They can also be proved without direct calculation.

pro

748. First Property.-The determinant ▲ is the sum of the ducts of each element of any column or row by its corresponding minor determinant, i. e.,

(6)

1 1

According to the definition of minor determinants, a,4, is the algebraic sum of all the terms of the determinant A which contain the factor a; b,B, is the algebraic sum of all terms which contain the factor b1 and c, C, of all terms which contain the factor c1. Hence (i) every term of the determinant is contained in the sum (6), for each term contains an element of the first row, hence either a, or b, or c1; and (ii) no term is counted twice in this sum, for no term contains two elements of the first row, i. e., does not contain at the same time a and b, or a, and e1, or b and c,. In order to prove that ▲ = a¡Å ̧ + ɑ‚Â1⁄2 + 9,4, it is necessary and sufficient to prove that every term of the determinant contains but one element of the first column of A.

3

749. Second Property. any row or column by the minor determinant which belongs to the corresponding element of a parallel row or column, is zero, that is

The sum of the products of each element of

Substitute in the determinant A the elements a b

of the third row for the elements a, b, c, of the first row, then the minor determinants with respect to the elements of the first column will remain the same (746, Cor.) A1, B1, C'1. The new determinant ▲,

This determinant = 0, because the first and third rows are the same (745). Therefore

one may use in a similar manner the theorem that a determinant with two equal columns = 0.

V. SOLUTION OF EQUATIONS OF THE FIRST DEGREE IN THREE UNKNOWN QUANTITIES

750. The Elimination of Two Unknown Quantities from Three Equations of the First Degree.

Determine the conditions under which the three equations

ax + by = cz

or

cz = 0,

are compatible with each other (233).

Put A = (abc) and represent the minor determinants of ▲ by A1, B1, C1, A1, etc. Multiply (1) by C1, (2) by C, (3) by C, and add the equations; then we have

3

(4) (Ca+Ca2+C3A3)x+(C{}},+C12+C};)y−(C ̧¢ ̧+C‚¢2+C ̧¢ ̧)=0.

But according to the properties of minor determinants the coefficients of x and y are zero (2749); hence from (4)

From equations (1), (2), (3) we have deduced equation (4), which is the necessary condition that (1), (2), (3) are compatible.

Conversely, if this condition exists, i. e., if equation (4) or (5) is true, then any equation of (1), (2), (3) can be deduced from the other two; hence this condition is sufficient.

The determinant A is called the eliminant and ▲ =0 the resultant of the system of equations (1), (2), (3), or of the equivalent system of homogeneous equations

successively in equations (1), (2), (3).

Hence, the eliminant of three equations of the first degree, (1), (2), (3), is the determinant of the coefficients of the unknown and the known terms, or of the coefficients of the equations written in homogeneous form, (1'), (2′), (3′). The resultant is found by placing ▲ = 0.

3

751. In case C1 = C1 = C = 0, equation (4) is satisfied, and hence the necessary condition for the compatibility of the equations is satisfied; but since equation (4), combined with two of the equations (1), (2), (3), is not sufficient for the derivation of the third, no conclusion can be drawn in this case.

EXAMPLE.The equation of a straight line has the form

If the points (x, y), (x,y) lie on this line, the coördinates of these points must satisfy the equation y = ax + b, i. e.,

The elimination of a and b from these equations gives

which is the equation of a straight line passing through (x, y) (x, y).