Determinants and Their Applications in Mathematical Physics

2 1. Determinants, First Minors, and Cofactors

It follows from (i) and (ii) that

x 1 x 2 ···xn=

∑n

k 1 =

···

∑n

kn=

a 1 k 1 a 2 k 2 ···anknek 1 ek 2 ···ekn. (1.2.2)

When two or more of thek’s are equal,ek 1 ek 2 ···ekn= 0. When thek’s are

distinct, the productek 1 ek 2 ···ekncan be transformed into±e 1 e 2 ···enby

interchanging the dummy variableskrin a suitable manner. The sign of

each term is unique and is given by the formula

x 1 x 2 ···xn=





(n! terms)

∑

σna 1 k 1 a 2 k 2 ···ankn



e

1 e 2 ···en, (1.2.3)

where

σn= sgn

{

1234 ··· (n−1) n

k 1 k 2 k 3 k 4 ··· kn− 1 kn

}

(1.2.4)

and where the sum extends over alln! permutations of the numberskr,

1 ≤r ≤n. Notes on permutation symbols and their signs are given in

Appendix A.2.

The coefficient of e 1 e 2 ···en in (1.2.3) contains alln

2

elements aij,

1 ≤i, j ≤n, which can be displayed in a square array. The coefficient

is called a determinant of ordern.

Definition.

An=

∣ ∣ ∣ ∣ ∣ ∣ ∣

a 11 a 12 ··· a 1 n

a 21 a 22 ··· a 2 n

...................

an 1 an 2 ··· ann

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

=

(n! terms)

∑

σna 1 k 1 a 2 k 2 ···ankn. (1.2.5)

The array can be abbreviated to |aij|n. The corresponding matrix is

denoted by [aij]n. Equation (1.2.3) now becomes

x 1 x 2 ···xn=|aij|ne 1 e 2 ···en. (1.2.6)

Exercise.If

(

12 ··· n

j 1 j 2 ··· jn

)

is a fixed permutation, show that

An=|aij|n=

n! terms

∑

k 1 ,...,kn

sgn

(

j 1 j 2 ··· jn

k 1 k 2 ··· kn

)

aj 1 k 1 aj 2 k 2 ···ajnkn

=

n! terms

∑

k 1 ,...,kn

sgn

(

j 1 j 2 ··· jn

k 1 k 2 ··· kn

)

ak 1 j 1 ak 2 j 2 ···aknjn.