THIRD ORDER

535. If

(1) α1x + b1y + c1z = 0,

(2) a2x+by+ c22 = 0,

then

(3) αçx+by+ c22 = 0,

(4) {α1(b2c3 — b2€2) — а2(b1е3 − bgе1) + α3(b1€2 − b2€1) } x = 0, by eliminating y and z by the method of addition. This is quickly done by multiplying (1) by bc — bo2, (2) by − b13 + b2o1, and (3) by b12 – b11, and then adding the three results.

If x in (4) 0, then must

537. Prop. 1. The value of the determinant is not altered by changing the columns into rows, and the rows into columns.

538. From this it may be seen that a determinant can be reduced to the algebraic sum of a, b, c, times a deterc1 minant of the second order. These determinants of the second order are called Minors of the original determinant. a2 A2 C2 is the first minor, is the second minor, etc.

b2 c2
b3 C3
It is seen that the minor which is the coefficient of

; it is composed of the elements not found in the

same column and the same row in which a1 is found. It

is seen also that the same law holds true in the case of the To preserve the order of the letters,

minor to b1 and e

Explanation. From the sum of the products of the elements found on lines 1, 2, 3, in diagram No. 1, subtract the product of the elements found on lines 4, 5, 6.

+

Expansion

(-40)+(12)+(36) −(16)

— (15) — (— 72) = 49.

After a little practice the student

will not need to draw the arrows, or 57 +6 +2

even to repeat the first and second

rows of figures.

539. Prop. 2.

+3

Interchanging two adjacent columns or

rows of the determinant changes its sign but not its arith

.. (1)=(2) arithmetically, but with opposite signs.

540. Prop. 3. Multiplying each element in any column, or in any row, by the same number multiplies the determinant by that number.

541. Prop. 4.

When two columns or two rows of a determinant are identical, the value of the determinant is zero.