Page 267 - Advanced engineering mathematics

DEFINITION OF THE DETERMINANT
EVALUATION OF DETERMINANTS I
CHAPTER 8 EVALUATION OF DETERMINANTS II A
DETERMINANT FORMULA FOR A −1

Determinants

8.1 Definition of the Determinant

Determinants are scalars (numbers or sometimes functions) formed from square matrices accord-
ing to a rule we will develop. The Wronskian of two functions, seen in Chapter 2, is a
determinant, and we will shortly see determinants in other important contexts. This chapter
develops some properties of determinants that we will need to evaluate and make use of them.
Let n be an integer with n ≥ 2. A permutation of the integers 1,2,··· ,n is a rearrangement
of these integers. For example, if p is the permutation that rearranges

1,2,3,4,5,6 → 3,1,4,5,2,6,
then p(1) = 3, p(2) = 1, p(3) = 4, p(4) = 5, p(5) = 2 and p(6) = 6.

A permutation is characterized as even or odd according to a rule we will illustrate.
Consider the permutation
p : 1,2,3,4,5 → 2,5,1,4,3
of the integers 1,2,3,4,5. For each k in the permuted list on the right, count the number
of integers to the right of k that are smaller than k. There is one number to the right of 2
smaller than 2, three numbers to the right of 5 smaller than 5, no numbers to the right of 1
smaller than 1, one number to the right of 4 smaller than 4, and no numbers to the right of
3 smaller than 3. Since 1 + 3 + 0 + 1 + 0 = 5 is odd, p is an odd permutation. When this
sum is even, p is an even permutation.

If p is a permutation on 1,2,··· ,n, define

1 if p is an even permutation
σ(p) =
−1 if p is an odd permutation.
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October 14, 2010 14:26 THM/NEIL Page-247 27410_08_ch08_p247-266