Page 75 - A Course in Linear Algebra with Applications

3.1: The Definition of a Determinant 59

What are we to make of this expression? In the first place it
contains six terms, each of which is a product of three entries
of A. The second subscripts in each term correspond to the
six ways of ordering the integers 1, 2, 3, namely

1,2,3 2,3,1 3,1,2 2,1,3 3,2,1 1,3,2.

Also each term is a product of three entries of A, while three
of the terms have positive signs and three have negative signs.
There is something of a pattern here, but how can one
tell which terms are to get a plus sign and which are get a
minus sign? The answer is given by permutations.

Permutations
Let n be a fixed positive integer. By a permutation of
2
the integers 1, ,..., n we shall mean an arrangement of these
integers in some definite order. For example, as has been
observed, there are six permutations of the integers 1, 2, 3.
2
In general, a permutation of 1, ,..., n can be written in
the form

k , 12, ••• , in

where ii, i 2,. • •, i n are the integers 1, ,..., n in some order.
2
Thus to construct a permutation we have only to choose dis-
tinct integers ii, i 2, •.., i n from the set {1, ,..., n). Clearly
2
there are n choices for i\\ once i\ has been chosen, it cannot
be chosen again, so there are just n — 1 choices for i 2\ since
i\ and i 2 cannot be chosen again, there are n — 2 choices for
^3, and so on. There will be only one possible choice for i n
since n — 1 integers have already been selected. The number
of ways of constructing a permutation is therefore equal to the
product of these numbers

n(n - l)(n - 2) •• 2-1,
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