Page 297 - Determinants and Their Applications in Mathematical Physics

282 6. Applications of Determinants in Mathematical Physics

6.9.2 Three Determinants
The determinant A and its cofactors are closely related to the Matsuno
determinant E and its cofactor (Section 5.4)

A = K n E,
2c r A rs = K n E rs ,
4c r c s A rs,rs = K n E rs,rs ,
where

n

K n =2 n c r . (6.9.4)
r=1
The proofs are elementary. It has been proved that

n n n

E rr = E rs ,
r=1 r=1 s=1
n n n n

E rs,rs = −2 † c s E rs .
r=1 s=1 r=1 s−1
It follows that

n n n

c r A rr = c r A rs (6.9.5)
r=1 r=1 s=1
n n n n

†
c r c s A rs,rs = − c r c s A rs . (6.9.6)
r=1 s=1 r=1 s=1
Define the determinant B as follows:
B = |b ij | n ,
where
a ij − 1


c i +c j
b ij = , j = i (6.9.7)
c i −c j
 2
ωθ i , j = i (ω = −1).
It may be verified that, for all values of i and j,
b ji = −b ij , j = i,
b ij − 1= −a ,
∗
ji
∗
a − 1= −b ji . (6.9.8)
ij
∗
When j = i, a = a ij , etc.
ij
Notes on bordered determinants are given in Section 3.7. Let P denote
the determinant of order (n + 2) obtained by bordering A by two rows and