146 4. Particular Determinants
Proof. Apply the Jacobi identity (Section 3.6.1) toAr, wherer≥i+1:
∣
∣
∣
∣
∣
A
(r)
ij
A
(r)
ir
A
(r)
rj A
(r)
rr
∣
∣
∣
∣
∣
=ArA
(r)
ir,jr
,
=ArA
(r−1)
ij
,
Ar− 1 A
(r)
ij
−ArA
(r−1)
ij
=A
(r)
ir
A
(r)
jr
. (4.11.47)
Scale the cofactors and refer to Theorems 4.48 and 4.49a:
A
ij
r−A
ij
r− 1
=
Ar
Ar− 1
(
A
ri
r A
rj
r
)
=2
−(4r−5)
A
ri
rA
rj
r
=2λr− 1 ,i− 1 λr− 1 ,j− 1. (4.11.48)
Hence,
2
n
∑
r=i+1
λr− 1 ,i− 1 λr− 1 ,j− 1 =
n
∑
r=i+1
(
A
ij
r
−A
ij
r− 1
)
=A
ij
n−A
ij
i
=A
ij
n
− 2
2(i−1)
λi− 1 ,j− 1 , (4.11.49)
which yields a formula for the scaled cofactorA
ij
n. The stated formula for
the simple cofactorA
(n)
ij follows from Theorem 4.49a.
Let
En=|Pm(0)|n, 0 ≤m≤ 2 n− 2 , (4.11.50)
wherePm(x) is the Legendre polynomial [Appendix A.5]. Then,
P 2 m+1(0)=0,
P 2 m(0) = νm. (4.11.51)
Hence,
En=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
ν 0 • ν 1 • ν 2 ···
- ν 1 • ν 2 • ···
ν 1 • ν 2 • ν 3 ···
- ν 2 • ν 3 • ···
ν 2 • ν 3 • ν 4 ···
........................
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n
. (4.11.52)
Theorem 4.51.
En=|Pm(0)|n=(−1)
n(n−1)/ 2
2
−(n−1)
2
.