752. The Solution of a System of Three Equations of the First Degree in Three Unknown Quantities when the Determinant of the Coefficients is not

0.

Multiply the equations in (1) respectively by A, A, A, and add

The coefficients of y and z are zero (2749); therefore

Similarly, multiply the equations in (1) respectively by B1, B, By and again by C, C, C, and add; whence we get

The determinant of x, y, z is the determinant whose elements are the coefficients of the equations; if in this determinant the coefficients of the several equations are replaced respectively by the second members of the equations, one obtains the numerator of x, and in a similar manner the numerator of y and of z.

NOTE 1.-Here it is clear that the denominator, A, of the value of x, y, and z should not be zero, for then the values of x, y, and z would be in case the numerators were not zero, and indeterminate in case they were zero.

NOTE 2.-That the values for x, y, and z are correct can be tested by substituting them in one or more of the equations in (1) and verifying the identities.

2

1

3

If d1 = d2 = d1 = 0 in equations (1), 752, i. e., if these three equations are homogeneous, then in case ▲ is not zero the only solution these equations have is x = y = z = 0.

Here Wis supposed to be not zero. Hence we will have from (2) and (3), 752, after simple transformations,

1

d2

d3

-Y, Z, W are proportional to the

Hence it follows, that X, Y, Z,

determinants, which are found by striking out the first, second, third, and fourth columns of the table,

754. Fifth Property. The elements of a row or column can be multiplied by any number and added to or subtracted from the corresponding elements of any other row or column of a determinant without altering the value of the determinant; thus we have

Proof. The first determinant is, according to 2747, equal to

(a1+mb1—ne̟ ̧)Ã ̧+(a2+mb,—nc„) Â ̧+(a ̧+mb ̧−nc ̧)4=

3

3

2 2

3

But a14, + a24,+ a ̧Â, is, according to the first property of determinants, 747, equal to A. The parentheses b14 ̧ + b„Ä + b,4, and c111+ c2A2+ €,4, are, according to the second property of determinants, 8748, equal to zero. Since rows and columns can be interchanged without altering the value of the determinant, the theorem applies to rows as well as to columns. This is a very important principle in the reduction of determinants.

For example:

The second determinant is derived from the first by multiplying the elements of the third column by 2 and subtracting them from the first column, and adding the elements of the third column to the elements of the second column; the third determinant is derived from the second determinant by adding the first column of the second determinant to the second column, and subtracting twice the elements of the first column from the third column. in the reduction follows from 1747.

The next step

|1b+a]

(b-a) (c-a)

1 cta

|=(a−b) (a−c)(c+a—a—b)=(a−b) (b−c)(c—a).

First step, subtract the first row from the second and third rows; second step, 747; third step, 741; fourth step, 734.

REMARK. The principle of addition can be used to give a determinant of the second order the form of a determinant of the third order. Thus we have

Calculate the values of p, q, r, s and prove that q2 = p (r+s), if