Page 18 - 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

y i = x i − a ij e j
=(a i1 e 1 + ··· + a i,j−1 e j−1 )+(a i,j+1 e j+1 + ··· + a in e n ) (1.3.1)
n−1

= a e , (1.3.2)

ik k
k=1
where

1 ≤ k ≤ j − 1
e = e k
k
= e k+1 , j ≤ k ≤ n − 1 (1.3.3)
a 1 ≤ k ≤ j − 1
ik = a ik
= a i,k+1 , j ≤ k ≤ n − 1. (1.3.4)
Note that each a is a function of j.
ik
It follows from Identity (ii) that
y 1 y 2 ··· y n = 0 (1.3.5)
since each y r is a linear combination of (n − 1) vectors e k so that each of
the (n − 1) terms in the expansion of the product on the left contains at
n
least two identical e’s. Referring to (1.3.1) and Identities (i) and (ii),

x 1 ··· x i−1 e j x i+1 ··· x n
=(y 1 + a 1j e j )(y 2 + a 2j e j ) ··· (y i−1 + a i−1,j e j )
e j (y i+1 + a i+1,j e j ) ··· (y n + a nj e j )
(1.3.6)
= y 1 ··· y i−1 e j y i+1 ··· y n
=(−1) n−i (y 1 ··· y i−1 y i+1 ··· y n )e j . (1.3.7)
From (1.3.2) it follows that

y 1 ··· y i−1 y i+1 ··· y n = M ij (e e ··· e ), (1.3.8)
1 2 n−1
where

M ij = σ n−1 a a ··· a a ··· a (1.3.9)
1k 1 2k 2 i−1,k i−1 i+1,k i+1 n−1,k n−1
and where the sum extends over the (n − 1)! permutations of the numbers
1, 2,..., (n − 1). Comparing M ij with A n , it is seen that M ij is the deter-
minant of order (n − 1) which is obtained from A n by deleting row i and
column j, that is, the row and column which contain the element a ij . M ij
is therefore associated with a ij and is known as a first minor of A n .
Hence, referring to (1.3.3),

x 1 ··· x i−1 e j x i+1 ··· x n
=(−1) n−i M ij (e e ··· e

1 2 n−1 )e j