Page 78 - A Course in Linear Algebra with Applications

62 Chapter Three: Determinants

Theorem 3.1.3

If n > 1, there are \{n\) even permutations of 1,2,... ,n and
the same number of odd permutations.

Proof
If the first two integers are interchanged in a permutation,
it is clear from the crossover diagram that an inversion is
either added or removed. Thus the operation changes an even
permutation to an odd permutation and an odd permutation
to an even one. This makes it clear that the numbers of even
and odd permutations must be equal. Since the total number
of permutations is n\, the result follows.

Example 3.1.2

The even permutations of 1, 2, 3 are

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

while the odd permutations are

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

Next we define the sign of a permutation i\, i2,. •., i n
sign(ii, i 2 ,..., i n )

to be +1 if the permutation is even and —1 if the permutation
is odd. For example, sign(3, 2, 1) = —1 since 3, 2, 1 is an odd
permutation.
Permutation matrices

Before proceeding to the formal definition of a determi-
nant, we pause to show how permutations can be represented
by matrices. An nxn matrix is called a permutation matrix if
it can be obtained from the identity matrix I n by rearranging
the rows or columns. For example, the permutation matrix