Page:Elementary algebra (1896).djvu/465

447 Rh 555. Signs of the Terms. Although we may always develop a determinant by means of the process described above, it is not always the simplest method, especially when our object is not so much to find the value of the whole determinant, as to find the signs of its several elements.

556. The expanded form of the determinant

a_1 b_1 c_1 a_2 b_2 c_2 a_3 b_3 c_3

$$=a_1b_2c_3 - a_1b_3c_2 + a_2b_3c_1 - a_2b_1c_3 + a_3b_1c_2 - a_3b_2c_1$$ ;

and it appears that each element is the product of three factors, one taken from each row, and one from each column; also the signs of half the terms are + and of the other half -. When written as above the signs of the several elements may be obtained as follows. The first element $$a_1b_2c_3$$; in which the suffixes follow the arithmetical order, is positive; we shall call this the leading element; every other element may he obtained from it by suitably interchanging the suffixes. The sign + or - is to be prefixed to any element according as the number of inversions of order in the line of suffixes is even or odd; for instance in the element $$a_3b_2c_1$$, 2 and 1 are out of their natural order, or inverted with respect to 3; 1 is inverted with respect to 2; hence there are three inversions and the sign of the element is negative ; in the element $$a_3b_1c_2$$, there are two inversions, hence the sign is positive.

557. The determinant whose leading element is $$a_1b_2c_3d_4\ldots$$ may thus be expressed by the notation

$$\sum \pm a_1b_2c_3d_4\ldots$$,

the $$\sum \pm$$ placed before the leading element indicating the aggregate of all the elements which can be obtained from it by suitable interchanges of suffixes and adjustment of signs. Sometimes the determinant is still more simply expressed by enclosing the leading element within brackets; thus ($$a_1b_2c_3d_4\ldots$$) is used as an abbreviation of $$\sum \pm a_1b_2c_3d_4\ldots$$.