Page 17 - Determinants and Their Applications in Mathematical Physics

2 1. Determinants, First Minors, and Cofactors

It follows from (i) and (ii) that

n n

e e
a
x 1 x 2 ··· x n = ··· a 1k 1 2k 2 ··· a nk n k 1 k 2 ··· e k n . (1.2.2)
k 1 =1 k n =1
e = 0. When the k’s are
When two or more of the k’s are equal, e k 1 k 2 ··· e k n
e can be transformed into ±e 1 e 2 ··· e n by
distinct, the product e k 1 k 2 ··· e k n
interchanging the dummy variables k r in a suitable manner. The sign of
each term is unique and is given by the formula
 
(n! terms)

x 1 x 2 ··· x n =  σ n a 1k 1 2k 2 ··· a nk n  e 1 e 2 ··· e n , (1.2.3)
a
where

1 2 3 4 ··· (n − 1) n
σ n = sgn (1.2.4)
k 1 k 2 k 3 k 4 ··· k n−1 k n
and where the sum extends over all n! permutations of the numbers k r ,
1 ≤ r ≤ n. Notes on permutation symbols and their signs are given in
Appendix A.2.
2
The coefficient of e 1 e 2 ··· e n in (1.2.3) contains all n elements a ij ,
1 ≤ i, j ≤ n, which can be displayed in a square array. The coefficient
is called a determinant of order n.

Definition.

a 11 a 12 ··· a 1n
(n! terms)

a 21 a 22 ··· a 2n
= a . (1.2.5)
A n = σ n a 1k 1 2k 2 ··· a nk n
...................

a n1 a n2 ··· a nn n
The array can be abbreviated to |a ij | n . The corresponding matrix is
denoted by [a ij ] n . Equation (1.2.3) now becomes
x 1 x 2 ··· x n = |a ij | n e 1 e 2 ··· e n . (1.2.6)

1 2 ··· n
Exercise. If is a fixed permutation, show that
j 1 j 2 ··· j n
n! terms
j 1 j 2 ···
A n = |a ij | n = sgn j n a j 1 k 1 j 2 k 2 ··· a j n k n
a
k 1 k 2 ··· k n
k 1 ,...,k n
n! terms
j 1 j 2 ···
a
= sgn j n a k 1 j 1 k 2 j 2 ··· a k n j n .
k 1 k 2 ··· k n
k 1 ,...,k n