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