Determinants and Their Applications in Mathematical Physics

3.2 Second and Higher Minors and Cofactors 19

called arejecterminor. The numbersisandjsare known respectively as

row and column parameters.

Now, letNi

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

denote the subdeterminant of orderrwhich is

obtained fromAnbyretainingrowsi 1 ,i 2 ,...,irand columnsj 1 ,j 2 ,...,jr

and rejecting the other rows and columns.Ni 1 i 2 ...ir;j 1 j 2 ...jr may conve-

niently be called aretainerminor.

Examples.

M

(5)

13 , 25

=

∣ ∣ ∣ ∣ ∣ ∣

a 21 a 23 a 24

a 41 a 43 a 44

a 51 a 53 a 54

∣ ∣ ∣ ∣ ∣ ∣

=N 245 , 134 ,

M

(5)

245 , 134

=

∣

∣

∣

∣

a 12 a 15

a 32 a 35

∣

∣

∣

∣

=N 13 , 25.

The minorsM

(n)

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

andNi

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

are said to be mutually

complementary inAn, that is, each is the complement of the other inAn.

This relationship can be expressed in the form

M

(n)

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

Ni

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

= compM

(n)

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

. (3.2.1)

The order and structure of rejecter minors depends on the value ofnbut

the order and structure of retainer minors are independent ofnprovided

only thatnis sufficiently large. For this reason, the parameternhas been

omitted fromN.

Examples.

Nip=

∣

∣a

ip

∣

∣

1

=aip,n≥ 1 ,

Nij,pq=

∣

∣

∣

∣

aip aiq

ajp ajq

∣

∣

∣

∣

,n≥ 2 ,

Nijk,pqr=

∣ ∣ ∣ ∣ ∣ ∣

aip aiq air

ajp ajq ajr

akp akq akr

∣ ∣ ∣ ∣ ∣ ∣

,n≥ 3.

Both rejecter and retainer minors arise in the construction of the Laplace

expansion of a determinant (Section 3.3).

Exercise.Prove that
∣
∣
∣
∣

Nij,pq Nij,pr

Nik,pq Nik,pr

∣

∣

∣

∣

=NipNijk,pqr.

3.2.2 Second and Higher Cofactors.............

The first cofactorA

(n)

ij

is defined in Chapter 1 and appears in Chapter 2.

It is now required to generalize that concept.