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)