Page 298 - Determinants and Their Applications in Mathematical Physics

6.9 The Benjamin–Ono Equation 283

two columns as follows:

c 1
1

c 2
1

··· ···

(6.9.9)
[a ij ] n ··· ···

P =
··· ···

c n 1
0 0

−1 − 1 ··· − 1 0 0
−c 1 − c 2 ··· − c n
n+2
and let Q denote the determinant of order (n + 2) obtained by bordering
B in a similar manner. Four of the cofactors of P are

1

1

···
, (6.9.10)

P n+1,n+1 =
···
[a ij ] n

···

1
−1 − 1 ··· − 1 0

n+1

c 1

c 2

···
[a ij ] n ···
P n+1,n+2 = −

···

c n
−1 − 1 ··· − 1 0

n+1

= c r A rs , (6.9.11)
r s

1

1

···
, (6.9.12)

P n+2,n+1 = −
···
[a ij ] n

···

1
0

n+1
−c 1 − c 2 ··· − c n

c 1

c 2

···

P n+2,n+2 = [a ij ] n ···

···

c n
0

n+1
−c 1 − c 2 ··· − c n