Determinants and Their Applications in Mathematical Physics

62 4. Particular Determinants

referring to Lemma (a) withn→s+ 1 and Lemma (c) withm→s+1,

s+1

∏

r=1

Mr=









Vn

∏s

r=1

(xs+1−xr)

∏n

r=s+2

(xr−xs+1)









V(xs+1,xs+2,...,xn)V

s− 1

n

V(x 1 ,x 2 ,...,xs)

=

V

s

n

[

V(x 1 ,x 2 ,...,xs)

∏s

r=1

(xs+1−xr)

]









V(xs+1,xs+2,...,xn)

∏n

r=s+2

(xr−xs+1)









=

V(xs+2,xs+3,...,xn)V

s

n

V(x 1 ,x 2 ,...,xs+1)

.

Hence, (a) is valid whenm=s+ 1, which proves (a). To prove (b), put

m=n−1 in (a) and useMn=Vn− 1. The details are elementary.

The proof of (c) is obtained by applying the permutation

{

123 ··· n

k 1 k 2 k 3 ··· kn

}

to (a). The only complication which arises is the determination of the sign

of the expression on the right of (c). It is left as an exercise for the reader

to prove that the sign is positive.

Exercise.LetA 6 denote the determinant of order 6 defined in column

vector notation as follows:

Cj=

[

ajajxjajx

2

j

bjbjyjbjy

2

j

]T

, 1 ≤j≤ 6.

Apply the Laplace expansion theorem to prove that

A 6 =

∑

i<j<k

p<q<r

σaiajakbpbqbrV(xi,xj,xk)V(yp,yq,yr),

where

σ= sgn

{

123456

ijkpqr

}

and where the lower set of parameters is a permutation of the upper set.

The number of terms in the sum is

(

6

3

)

= 20.

Prove also that

A 6 = 0 when aj=bj, 1 ≤j≤ 6.

Generalize this result by giving an expansion formula forA 2 nfrom the

firstmrows and the remaining (2n−m) rows using the dummy variables

kr,1≤r≤ 2 n. The generalized formula and Theorem (c) are applied in

Section 6.10.4 on the Einstein and Ernst equations.