Determinants and Their Applications in Mathematical Physics

18 3. Intermediate Determinant Theory

Exercises

1.Letδ

r

denote an operator which, when applied toCj, has the effect

of dislocating the elementsrpositions downward in a cyclic manner

so that the lowest set ofrelements are expelled from the bottom and

reappear at the top without change of order.

δ

r

Cj=

[

an−r+1,jan−r+2,j···anja 1 ja 2 j···an−r,j

]T

,

1 ≤r≤n− 1 ,

δ

0

Cj=δ

n

Cj=Cj.

Prove that

n

∑

j=1

∣

∣

C 1 ···δ

r

Cj···Cn

∣

∣

=

{

0 , 1 ≤r≤n− 1

nA, r=0,n.

2.Prove that

n

∑

r=1

∣

∣C

1 ···δ

r

Cj···Cn

∣

∣=s

jSj,

where

sj=

n

∑

i=1

aij,

Sj=

n

∑

i=1

Aij.

Hence, prove that an arbitrary determinant An = |aij|n can be

expressed in the form

An=

1

n

n

∑

j=1

sjSj. (Trahan)

3.2 Second and Higher Minors and Cofactors..........

3.2.1 Rejecter and Retainer Minors

It is required to generalize the concept of first minors as defined in

Chapter 1.

LetAn=|aij|n, and let{is}and{js},1≤s≤r≤n, denote two

independent sets ofr distinct numbers, 1 ≤ is andjs ≤n. Now let

M

(n)

i 1 i 2 ...ir;j 1 j 2 ...jr denote the subdeterminant of order (n−r) which is ob-

tained fromAnbyrejectingrowsi 1 ,i 2 ,...,irand columnsj 1 ,j 2 ,...,jr.

M

(n)

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

is known as anrth minor ofAn. It may conveniently be