Page 64 - Determinants and Their Applications in Mathematical Physics

3.7 Bordered Determinants 49

Proof. It follows from (3.7.2) that

n n

A ir,js =0, 1 ≤ i, r ≤ n.
j=1 s=1
Expanding B ij by elements from the last column,
n n

B ij = − x r A ir,js .
r=1 s=1
Hence

n n n n

B ij = − x r A ir,js
j=1 r=1 j=1 s=1
=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
n

···
= u p v q x r y s A pq,rs ,
[a ij ] n ···

u n v n
p,q,r,s=1

• •
x 1 x 2 ··· x n

• •
n+2
y 1 y 2 ··· y n
where
A = |a ij | n .
Proof. Denote the determinant by B and apply the Jacobi identity to
cofactors obtained by deleting one of the last two rows and one of the last
two columns

BB n+1,n+2;n+1,n+2
B n+1,n+1

= (3.7.3)
B n+1,n+2

BA.
B n+2,n+1 B n+2,n+2

Each of the first cofactors is a determinant with single borders

v 1

v 2

B n+1,n+1 = [a ij ] n ···

v n

•
n+1
y 1 y 2 ··· y n