Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

DETERMINANTS

365

from each column of the square array. Moreover, every product
of this type is present as some term in the determinant, as can
be shown by writing down all such products and comparing
them with the terms of the determinant.
By analogy, then, to form the determinant a b c d, we
should write down all the products of elements of (1), each of
which contains just one factor from each row and just one
factor from each column of (1), that is, all the products of the
form aibjckdl, where i, j, k, I are the numbers 1, 2, 3, 4 in all
possible orders. There are 24 such products. For, we can
choose the first factor, say from the column of a's, in four ways
- from any one of the four rows; and then the second factor,
say from the column of b's, in three ways - from any one of the
three remaining rows; and the third factor, in two ways; the
fourth is then uniquely determined. The number of possible
products is, therefore, 4 * 3 2 1 = 4! = 24.
It remains to determine the signs to be given to the 24
products. Toward this end, let us write down the subscripts of
the terms of (4), ~ 2, in the order in which they occur, when
the letters a, b, c are in their natural order. For the terms with
plus signs we have
12 3, 2 3 1, 31 2,
and for the terms with minus signs,
321, 21 3, 132.
The first set 1 2 3 is normal. In the second set, 2 3 1, 2 and
3 each precede 1, and we say that there are two inversions from
the normal order. In 3 1 2, 3 precedes 1 and 2, -again two
inversions. In the three sets for the negative terms the number of inversions is respectively three, one, and one.
It appears, then, that the number of inversions in the set of subscripts for a term with a plus sign is even (or zero), whereas for a
term with a minus sign, this number is always odd.
Proceeding according to this rule, we should give to each of
the 24 products, abjcd,, formed from (1) a plus sign or a minus