Determinants and Their Applications in Mathematical Physics

3.7 Bordered Determinants 49

Proof. It follows from (3.7.2) that

n

∑

j=1

n

∑

s=1

Air,js=0, 1 ≤i, r≤n.

ExpandingBijby elements from the last column,

Bij=−

n

∑

r=1

xr

n

∑

s=1

Air,js.

Hence

n

∑

j=1

Bij=−

n

∑

r=1

xr

n

∑

j=1

n

∑

s=1

Air,js

=0.

Bordered determinants appear in other sections including Section 4.10.3

on the Yamazaki–Hori determinant and Section 6.9 on the Benjamin–Ono

equation.

3.7.2 A Determinant with Double Borders

Theorem 3.13.

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

u 1 v 1

u 2 v 2

[aij]n ··· ···

un vn

x 1 x 2 ···xn ••

y 1 y 2 ···yn ••

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+2

=

n

∑

p,q,r,s=1

upvqxrysApq,rs,

where

A=|aij|n.

Proof. Denote the determinant byBand apply the Jacobi identity to

cofactors obtained by deleting one of the last two rows and one of the last

two columns

∣

∣

∣

∣

Bn+1,n+1 Bn+1,n+2

Bn+2,n+1 Bn+2,n+2

∣

∣

∣

∣

=

BBn+1,n+2;n+1,n+2

BA.

(3.7.3)

Each of the first cofactors is a determinant with single borders

Bn+1,n+1=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

v 1

v 2

[aij]n ···

vn

y 1 y 2 ···yn •

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1