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