Determinants and Their Applications in Mathematical Physics

3.2 Second and Higher Minors and Cofactors 21

is possible to expandA

(n)

ip by elements from any row or column and second

cofactorsA

(n)

ij,pq

. The formula for row expansions is

A

(n)

ip

=

n

∑

q=1

ajqA

(n)

ij,pq

, 1 ≤j≤n, j=i. (3.2.3)

The term in whichq=pis zero by the first convention for cofactors. Hence,

the sum contains (n−1) nonzero terms, as expected. The (n−1) values of

jfor 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 parameternand referring to (2.3.10), it follows that ifi<j

andp<q, then

Aij,pq=

∂Aip

∂ajq

=

∂

2

A

∂aip∂ajq

(3.2.4)

which can be regarded as an alternative definition of the second cofactor

Aij,pq.

Similarly,

A

(n)

ij,pq

=

n

∑

r=1

akrA

(n)

ijk,pqr

, 1 ≤k≤n, k=iorj. (3.2.5)

Omitting the parametern, it follows that ifi<j<kandp<q<r, then

Aijk,pqr=

∂Aij,pq

∂akr

=

∂

3

A

∂aip∂ajq∂akr

(3.2.6)

which can be regarded as an alternative definition of the third cofactor

Aijk,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 anrth cofactor, a subdeterminant of order (n−r), can

be expressed in the form

A

(n)

i 1 i 2 ...ir;j 1 j 2 ...jr=

n

∑

q=1

apqA

(n)

i 1 i 2 ...irp;j 1 j 2 ...jrq, (3.2.7)

1 ≤p≤n, p=is, 1 ≤s≤r.

Therterms in whichq=js,1≤s≤r, are zero by the first convention

for cofactors. Hence, the sum contains (n−r) nonzero terms, as expected.