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
∣ ∣ ∣ ∣ ∣ ∣
∣
∣
∣
.
- 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,