Page 93 - Determinants and Their Applications in Mathematical Physics

78 4. Particular Determinants

Applying the Laplace expansion formula (Section 3.3) in reverse,

a 11 a 12 a 13

a 21 a 22 a 23

2 a 31 a 32 a 33
3 .
A =
−b 31 −b 32 −b 33 a 33 a 23 a 13

−b 21 −b 22 −b 23 a 32 a 22 a 12

−b 11 −b 12 −b 13 a 31 a 21 a 11
Now, perform the column and row operations

C = C j + C 7−j , 4 ≤ j ≤ 6,
j
R = R i + R 7−i , 1 ≤ i ≤ 3,

i
and show that the resulting determinant is skew-symmetric. Hence,
show that

A 3 = c 12 c 13 b 13 b 12 b 11

c 23 b 23 b 22 b 21
b 31 .

b 33 b 32

c 23 c 13

c 12
2. Theorem (Muir and Metzler) An arbitrary determinant of order 2n
can be expressed as a Pfaffian of order n.
Prove this theorem in the particular case in which n = 2 as follows:
Denote the determinant by A 4 , transpose it and interchange first rows
1 and 2 and then rows 3 and 4. Change the signs of the elements in the
(new) rows 2 and 4. These operations leave the value of the determinant
unaltered. Multiply the initial and final determinants together, prove
that the product is skew-symmetric, and, hence, prove that

A 4 = (N 12,12 + N 12,34 )(N 13,12 + N 13,34 )(N 14,12 + N 14,34 )

(N 23,12 + N 23,34 )(N 24,12 + N 24,34 ) .

(N 34,12 + N 34,34 )
where N ij,rs is a retainer minor (Section 3.2.1).
3. Expand Pf 3 by the five elements from the first row and their associated
second-order Pfaffians.
4. A skew-symmetric determinant A 2n is defined as follows:
A 2n = |a ij | 2n ,
where
a ij = x i − x j .
x i + x j
Prove that the corresponding Pfaffian is given by the formula

Pf 2n−1 = a ij ,
1≤i<j≤2n