derived from the primitive
arrangement only by an odd number, or else only by an even number of
interchanges,--a theorem the verification of which may be easily
obtained from the theorem (in fact a particular case of the general
one), an arrangement can be derived from itself only by an even number
of interchanges.] And this being so, each product has the sign belonging
to the corresponding arrangement of the columns; in particular, a
determinant contains with the sign + the product of the elements in its
dexter diagonal. It is to be observed that the rule gives as many
positive as negative arrangements, the number of each being = 1/2 1.2...n.

The rule of signs may be expressed in a different form. Giving to the
columns in the primitive arrangement the numbers 1, 2, 3 ... n, to
obtain the sign belonging to any other arrangement we take, as often as
a lower number succeeds a higher one, the sign -, and, compounding
together all these minus signs, obtain the proper sign, + or - as the
case may be.

Thus, for three columns, it appears by either rule that 123, 231, 312
are positive; 213, 321, 132 are negative; and the developed expression
of the foregoing determinant of the third order is

= ab'c" - ab"c' + a'b"c - a'bc" + a"bc' - a"b'c.

3. It further appears that a determinant is a linear function[1] of the
elements of each column thereof, and also a linear function of the
elements of each line thereof; moreover, that the determinant retains
the same value, only its sign being altered, when any two columns are
interchanged, or when any two lines are interchanged; more generally,
when the columns are permuted in any manner, or when the lines are
permuted in any manner, the determinant retains its original value, with
the sign + or - according as the new arrangement (considered as derived
from the primitive arrangement) is positive or negative according to the
foregoing rule of signs. It at once follows that, if two columns are
identical, or if two lines are identical, the value of the determinant
is = 0. It may be added, that if the lines are converted into columns,
and the columns into lines, in such a way as to leave the dexter
diagonal unaltered, the value of the determinant is unaltered; the
determinant is in this case said to be _transposed_.

4. By what precedes it appears that there exists a function of the n squared
elements, linear as regards the terms of each column (or say, for
shortness, linear