Determinants and Their Applications in Mathematical Physics

230 5. Further Determinant Theory

Hence,

∣ ∣ ∣ ∣ ∣ ∣

Br+4 Br+3 Br+2

Br+3 Br+2 Br+1

Br+2 Br+1 Br

∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

A

(5)

12 , 12

A

(5)

12 , 15

A

(5)

12 , 45

A

(5)

15 , 12 A

(5)

15 , 15 A

(5)

15 , 45

A

(5)

45 , 12

A

(5)

45 , 15

A

(5)

45 , 45

∣ ∣ ∣ ∣ ∣ ∣ ∣

. (5.8.8)

Denote the determinant on the right byV 3. Then,V 3 is not a standard

third-order Jacobi determinant which is of the form

|A

(n)

pq|^3 or |A

(n)

gp,hq

| 3 ,p=i, j, k, q=r, s, t.

However,V 3 can be regarded as a generalized Jacobi determinant in which

the elements have vector parameters:

V 3 =|A

(5)

uv|^3 , (5.8.9)

whereuandv=[1,2], [1,5], and [4,5], andA

(5)

uv

is interpreted as a second

cofactor ofA 5. It may be verified that

V 3 =A

(5)

125;125

A

(5)

145;145

A 5 +φ 4 (A

(5)

15

)

2

(5.8.10)

and that if

V 3 =|A

(4)

uv|^3 , (5.8.11)

whereuandv=[1,2], [1,4], and [3,4], then

V 3 =A

(4)

124;124

A

(4)

134;134

A 4 +(A

(4)

14

)

2

. (5.8.12)

These results suggest the following conjecture:

Conjecture.If

V 3 =|A

(n)

uv

| 3 ,

whereuandv=[1,2], [1,n], and [n− 1 ,n], then

V 3 =A

(n)

12 n;12nA

(n)

1 ,n− 1 ,n;1,n− 1 ,nAn+A

(n)

12 ,n− 1 ,n;12,n− 1 ,n(A

(n)

1 n)

2

.

Exercise.If

V 3 =|A

(4)

uv

|,

where

u=[1,2],[1,3],and [2,4],

v=[1,2],[1,3],and [2,3].

prove that

V 3 =−φ 5 φ 6 A 4.