Determinants and Their Applications in Mathematical Physics

6.9 The Benjamin–Ono Equation 283

two columns as follows:

P=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

c 1 1

c 2 1

··· ···

[aij]n ··· ···

··· ···

cn 1

−c 1 −c 2 ··· −cn 00

− 1 − 1 ··· − 100

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

∣

n+2

(6.9.9)

and letQdenote the determinant of order (n+ 2) obtained by bordering

Bin a similar manner. Four of the cofactors ofPare

Pn+1,n+1=

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1

1

···

[aij]n ···

···

1

− 1 − 1 ··· − 10

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

, (6.9.10)

Pn+1,n+2=−

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

c 1

c 2

···

[aij]n ···

···

cn

− 1 − 1 ··· − 10

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

=

∑

r

∑

s

crArs, (6.9.11)

Pn+2,n+1=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ ∣

1

1

···

[aij]n ···

···

1

−c 1 −c 2 ··· −cn 0

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ ∣

n+1

, (6.9.12)

Pn+2,n+2=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

c 1

c 2

···

[aij]n ···

···

cn

−c 1 −c 2 ··· −cn 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1