3.6 The Jacobi Identity and Variants 45
ButA
(n+1)
i,n+1;j,n+1=A
(n)
ij. Hence,AnP= 0. The result follows.
Three particular cases of (B) are required for the proof of the next
theorem.
Put (i, p, q)=(r, r, n), (n− 1 ,r,n), (n, r, n) in turn:
∣
∣
∣
∣
A
(n)
rr A
(n)
rn
A
(n+1)
n+1,r
A
(n+1)
n+1,n
∣
∣
∣
∣
−AnA
(n+1)
r,n+1;rn
=0, (B 1 )
∣
∣
∣
∣
∣
A
(n)
n− 1 ,r
A
(n)
n− 1 ,n
A
(n+1)
n+1,r
A
(n+1)
n+1,n
∣
∣
∣
∣
∣
−AnA
(n+1)
n− 1 ,n+1;rn
=0, (B 2 )
∣
∣
∣
∣
A
(n)
nr A
(n)
nn
A
(n+1)
n+1,r
A
(n+1)
n+1,n
∣
∣
∣
∣
−AnA
(n+1)
n,n+1;rn
=0. (B 3 )
Theorem 3.7.
∣ ∣ ∣ ∣ ∣ ∣ ∣
A
(n+1)
r,n+1;rn A
(n)
rr A
(n)
rn
A
(n+1)
n− 1 ,n+1;rn
A
(n)
n− 1 ,r
A
(n)
n− 1 ,n
A
(n+1)
n,n+1;rn A
(n)
nr A
(n)
nn
∣ ∣ ∣ ∣ ∣ ∣ ∣
=0.
Proof. Denote the determinant byQ. Then,
Q 11 =
∣
∣
∣
∣
∣
A
(n)
n− 1 ,r A
(n)
n− 1 ,n
A
(n)
nr A
(n)
nn
∣
∣
∣
∣
∣
=AnA
(n)
n− 1 ,n;rn
=AnA
(n−1)
n− 1 ,r
,
Q 21 =−AnA
(n−1)
rr
,
Q 31 =AnA
(n)
r,n−1;rn. (3.6.17)
Hence, expandingQby the elements in column 1 and applying (B 1 )–(B 3 ),
Q=An
[
A
(n+1)
r,n+1;rn
A
(n−1)
n− 1 ,r
−A
(n+1)
n− 1 ,n+1;rn
A
(n−1)
rr
+A
(n+1)
n,n+1;rn
A
(n)
r,n−1;rn
]
(3.6.18)
=A
(n−1)
n− 1 ,r
∣
∣
∣
∣
A
(n)
rr A
(n)
rn
A
(n+1)
n+1,r
A
(n+1)
n+1,n
∣
∣
∣
∣
−A
(n−1)
rr
∣
∣
∣
∣
∣
A
(n)
n− 1 ,r A
(n)
n− 1 ,n
A
(n+1)
n+1,r
A
(n+1)
n+1,n
∣
∣
∣
∣
∣
+A
(n)
r,n−1;rn
∣
∣
∣
∣
A
(n)
nr A
(n)
nn
A
(n+1)
n+1,r
A
(n+1)
n+1,n
∣
∣
∣
∣
=A
(n+1)
n+1,n
[
A
(n)
nrA
(n)
r,n−1;rn−
∣
∣
∣
∣
A
(n−1)
rr A
(n)
rr
A
(n−1)
n− 1 ,r A
(n)
n− 1 ,r
∣
∣
∣
∣
]
−A
(n+1)
n+1,r
[
An− 1 A
(n)
r,n−1;rn−
∣
∣
∣
∣
A
(n−1)
rr A
(n)
rn
A
(n−1)
n− 1 ,r A
(n)
n− 1 ,n