Page 36 - Determinants and Their Applications in Mathematical Physics

3.2 Second and Higher Minors and Cofactors 21
(n)
is possible to expand A by elements from any row or column and second
ip
(n)
cofactors A . The formula for row expansions is
ij,pq
n
(n) (n)
A = a jq A , 1 ≤ j ≤ n, j = i. (3.2.3)
ip ij,pq
q=1
The term in which q = p is zero by the first convention for cofactors. Hence,
the sum contains (n − 1) nonzero terms, as expected. The (n − 1) values of
j for which the expansion is valid correspond to the (n − 1) possible ways
of expanding a subdeterminant of order (n − 1) by elements from one row
and their cofactors.
Omitting the parameter n and referring to (2.3.10), it follows that if i<j
and p<q, then

A ij,pq = ∂A ip
∂a jq
2
∂ A
= ∂a ip ∂a jq (3.2.4)
which can be regarded as an alternative definition of the second cofactor
A ij,pq .
Similarly,

n
(n) (n)
A = a kr A , 1 ≤ k ≤ n, k = i or j. (3.2.5)
ij,pq ijk,pqr
r=1
Omitting the parameter n, it follows that if i<j <k and p<q <r, then
A ijk,pqr = ∂A ij,pq
∂a kr
3
∂ A
= ∂a ip ∂a jq ∂a kr (3.2.6)
which can be regarded as an alternative definition of the third cofactor
A ijk,pqr .
Higher cofactors can be defined in a similar manner. Partial derivatives of
this type appear in Section 3.3.2 on the Laplace expansion, in Section 3.6.2
on the Jacobi identity, and in Section 5.4.1 on the Matsuno determinant.
The expansion of an rth cofactor, a subdeterminant of order (n−r), can
be expressed in the form

n
(n) (n)
A = a pq A , (3.2.7)
i 1 i 2 ...i r ;j 1 j 2 ...j r i 1 i 2 ...i r p;j 1 j 2 ...j r q
q=1
1 ≤ p ≤ n, p = i s , 1 ≤ s ≤ r.
The r terms in which q = j s ,1 ≤ s ≤ r, are zero by the first convention
for cofactors. Hence, the sum contains (n − r) nonzero terms, as expected.