Page 19 - Determinants and Their Applications in Mathematical Physics

4 1. Determinants, First Minors, and Cofactors

=(−1) n−i M ij (e ··· e )(e ··· e

1 j−1 j n−1 )e j
=(−1) n−i M ij (e 1 ··· e j−1 )(e j+1 ··· e n )e j
=(−1) i+j M ij (e 1 e 2 ··· e n ). (1.3.10)
Now, e j can be regarded as a particular case of x i as defined in (1.2.1):

n

e j = a ik e k ,
k=1
where

a ik = δ jk .
Hence, replacing x i by e j in (1.2.3),

x 1 ··· x i−1 e j x i+1 ··· x n = A ij (e 1 e 2 ··· e n ), (1.3.11)
where

a
A ij = σ n a 1k 1 2k 2 ··· a ik i ··· a nk n ,
where

=0 k i = j
a ik i
=1 k i = j.
Referring to the definition of a determinant in (1.2.4), it is seen that A ij is
the determinant obtained from |a ij | n by replacing row i by the row

[0 ... 010 ... 0],
where the element 1 is in column j. A ij is known as the cofactor of the
element a ij in A n .
Comparing (1.3.10) and (1.3.11),

A ij =(−1) i+j M ij . (1.3.12)
(n) (n)
Minors and cofactors should be written M and A but the parameter
ij ij
n can be omitted where there is no risk of confusion.
Returning to (1.2.1) and applying (1.3.11),

n

x 1 x 2 ··· x n = x 1 ··· x i−1 a ik e k x i+1 ··· x n
k=1
n

= a ik (x 1 ··· x i−1 e k x i+1 ··· x n )
k=1

n

= a ik A ik e 1 e 2 ··· e n . (1.3.13)
k=1