Page 245 - Determinants and Their Applications in Mathematical Physics

230 5. Further Determinant Theory

Hence,
(5) (5) (5)
A A A

12,12 12,15
B r+4 B r+3 12,45
B r+2 (5) (5) (5)
= A A A . (5.8.8)
B r+3 B r+2
15,12 15,15 15,45
B r+1

(5) (5) (5)
B r+2 B r+1
A A A
B r
45,12 45,15 45,45
Denote the determinant on the right by V 3 . Then, V 3 is not a standard
third-order Jacobi determinant which is of the form
|A (n) | or |A (n) | p = i, j, k, q = r, s, t.
gp,hq 3 ,
pq 3
However, V 3 can be regarded as a generalized Jacobi determinant in which
the elements have vector parameters:
(5)
V 3 = |A uv 3 , (5.8.9)
|
(5)
where u and v =[1, 2], [1, 5], and [4, 5], and A uv is interpreted as a second
cofactor of A 5 . It may be verified that
(5) (5) (5) 2
V 3 = A 125;125 A 145;145 A 5 + φ 4 (A 15 ) (5.8.10)
and that if
(4)
V 3 = |A uv 3 , (5.8.11)
|
where u and v =[1, 2], [1, 4], and [3, 4], then
(4) (4) (4) 2
V 3 = A A A 4 +(A ) . (5.8.12)
124;124 134;134 14
These results suggest the following conjecture:
Conjecture. If
(n)
V 3 = |A uv 3 ,
|
where u and v =[1, 2], [1,n], and [n − 1,n], then
(n) (n) (n) (n) 2
V 3 = A A A n + A (A ) .
12n;12n 1,n−1,n;1,n−1,n 12,n−1,n;12,n−1,n 1n
Exercise. If
(4)
V 3 = |A 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 .