Determinants and Their Applications in Mathematical Physics

78 4. Particular Determinants

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

A

2

3

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

a 11 a 12 a 13

a 21 a 22 a 23

a 31 a 32 a 33

−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

′

j=Cj+C^7 −j,^4 ≤j≤^6 ,

R

′

i=Ri+R^7 −i,^1 ≤i≤^3 ,

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 33 b 32 b 31

c 23 c 13

c 12

∣ ∣ ∣ ∣ ∣ ∣

∣

∣

∣

.

2. Theorem (Muir and Metzler)An arbitrary determinant of order 2 n

can be expressed as a Pfaffian of ordern.

Prove this theorem in the particular case in whichn= 2 as follows:

Denote the determinant byA 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 )

∣ ∣ ∣ ∣ ∣ ∣

.

whereNij,rsis 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 determinantA 2 nis defined as follows:

A 2 n=|aij| 2 n,

where

aij=

xi−xj

xi+xj

.

Prove that the corresponding Pfaffian is given by the formula

Pf 2 n− 1 =

∏

1 ≤i<j≤ 2 n

aij,