Determinants and Their Applications in Mathematical Physics

22 3. Intermediate Determinant Theory

The (n−r) values ofpfor which the expansion is valid correspond to the

(n−r) possible ways of expanding a subdeterminant of order (n−r)by

elements from one row and their cofactors.

If one of the column parameters of anrth cofactor ofAn+1is (n+ 1),

the cofactor does not contain the elementan+1,n+1. If none of the row

parameters is (n+ 1), then therth cofactor can be expanded by elements

from its last row and their first cofactors. But first cofactors of anrth

cofactor ofAn+1are (r+ 1)th cofactors ofAn+1which, in this case, are

rth cofactors ofAn. Hence, in this case, anrth cofactor ofAn+1can be

expanded in terms of the firstnelements in the last row andrth cofactors

ofAn. This expansion is

A

(n+1)

i 1 i 2 ...ir;j 1 j 2 ...jr− 1 (n+1)

=−

n

∑

q=1

an+1,qA

(n)

i 1 i 2 ...ir;j 1 j 2 ...jr− 1 q

. (3.2.8)

The corresponding column expansion is

A

(n+1)

i 1 i 2 ...ir− 1 (n+1);j 1 j 2 ...jr

=−

n

∑

p=1

ap,n+1A

(n)

i 1 i 2 ...ir− 1 p;j 1 j 2 ...jr

. (3.2.9)

Exercise.Prove that

∂

2

A

∂aip∂ajq

=−

∂

2

A

∂aiq∂ajp

,

∂

3

A

∂aip∂ajq∂akr

=

∂

3

A

∂akp∂aiq∂ajr

=

∂

3

A

∂ajp∂akq∂air

without restrictions on the relative magnitudes of the parameters.

3.2.4 Alien Second and Higher Cofactors; Sum Formulas

The (n−2) elementsahq,1≤q≤n,q=horp, appear in the second

cofactorA

(n)

ij,pqifh=iorj. Hence,

n

∑

q=1

ahqA

(n)

ij,pq

=0,h=iorj,

since the sum represents a determinant of order (n−1) with two identical

rows. This formula is a generalization of the theorem on alien cofactors

given in Chapter 2. The value of the sum of 1≤h≤nis given by the sum

formula for elements and cofactors, namely

n

∑

q=1

ahqA

(n)

ij,pq=











A

(n)

ip,h=j=i

−A

(n)

jp

,h=i=j

0 , otherwise

(3.2.10)