Determinants and Their Applications in Mathematical Physics

4 1. Determinants, First Minors, and Cofactors

=(−1)

n−i

Mij(e

′

1

···e

′

j− 1

)(e

′

j

···e

′

n− 1

)ej

=(−1)

n−i

Mij(e 1 ···ej− 1 )(ej+1···en)ej

=(−1)

i+j

Mij(e 1 e 2 ···en). (1.3.10)

Now,ejcan be regarded as a particular case ofxias defined in (1.2.1):

ej=

n

∑

k=

aikek,

where

aik=δjk.

Hence, replacingxibyejin (1.2.3),

x 1 ···xi− 1 ejxi+1···xn=Aij(e 1 e 2 ···en), (1.3.11)

where

Aij=

∑

σna 1 k

1

a 2 k

2

···aik

i

···ank

n

,

where

aiki=0 ki=j

=1 ki=j.

Referring to the definition of a determinant in (1.2.4), it is seen thatAijis

the determinant obtained from|aij|nby replacing rowiby the row

[0... 010 ...0],

where the element 1 is in columnj.Aijis known as the cofactor of the

elementaijinAn.

Comparing (1.3.10) and (1.3.11),

Aij=(−1)

i+j

Mij. (1.3.12)

Minors and cofactors should be writtenM

(n)

ij

andA

(n)

ij

but the parameter

ncan be omitted where there is no risk of confusion.

Returning to (1.2.1) and applying (1.3.11),

x 1 x 2 ···xn=x 1 ···xi− 1

(

n

∑

k=

aikek

)

xi+1···xn

=

n

∑

k=

aik(x 1 ···xi− 1 ekxi+1···xn)

=

[

n

∑

k=

aikAik

]

e 1 e 2 ···en. (1.3.13)