Determinants and Their Applications in Mathematical Physics

1.3 First Minors and Cofactors 3

1.3 First Minors and Cofactors..................

Referring to (1.2.1), put

yi=xi−aijej

=(ai 1 e 1 +···+ai,j− 1 ej− 1 )+(ai,j+1ej+1+···+ainen) (1.3.1)

=

n− 1

∑

k=

a

′

ike

′

k, (1.3.2)

where

e

′

k

=ek 1 ≤k≤j− 1

=ek+1,j≤k≤n− 1 (1.3.3)

a

′

ik

=aik 1 ≤k≤j− 1

=ai,k+1,j≤k≤n− 1. (1.3.4)

Note that eacha
′
ik
is a function ofj.

It follows from Identity (ii) that

y 1 y 2 ···yn= 0 (1.3.5)

since eachyris a linear combination of (n−1) vectorsekso that each of

the (n−1)
n
terms in the expansion of the product on the left contains at

least two identicale’s. Referring to (1.3.1) and Identities (i) and (ii),

x 1 ···xi− 1 ejxi+1···xn

=(y 1 +a 1 jej)(y 2 +a 2 jej)···(yi− 1 +ai− 1 ,jej)

ej(yi+1+ai+1,jej)···(yn+anjej)

=y 1 ···yi− 1 ejyi+1···yn (1.3.6)

=(−1)

n−i

(y 1 ···yi− 1 yi+1···yn)ej. (1.3.7)

From (1.3.2) it follows that

y 1 ···yi− 1 yi+1···yn=Mij(e

′

1 e

′

2 ···e

′

n− 1 ), (1.3.8)

where

Mij=

∑

σn− 1 a

′

1 k 1

a

′

2 k 2

···a

′

i− 1 ,ki− 1

a

′

i+1,ki+

···a

′

n− 1 ,kn− 1

(1.3.9)

and where the sum extends over the (n−1)! permutations of the numbers

1 , 2 ,...,(n−1). ComparingMijwithAn, it is seen thatMijis the deter-

minant of order (n−1) which is obtained fromAnby deleting rowiand

columnj, that is, the row and column which contain the elementaij.Mij

is therefore associated withaijand is known as a first minor ofAn.

Hence, referring to (1.3.3),

x 1 ···xi− 1 ejxi+1···xn

=(−1)

n−i

Mij(e

′

1

e

′

2

···e

′

n− 1

)ej