3.6 The Jacobi Identity and Variants 45ButA
(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 nexttheorem.
Put (i, p, q)=(r, r, n), (n− 1 ,r,n), (n, r, n) in turn:∣
∣
∣
∣A
(n)
rr A(n)
rnA(n+1)
n+1,rA
(n+1)
n+1,n∣
∣
∣
∣
−AnA(n+1)
r,n+1;rn=0, (B 1 )
∣
∣
∣
∣
∣
A
(n)
n− 1 ,rA
(n)
n− 1 ,nA
(n+1)
n+1,rA
(n+1)
n+1,n∣
∣
∣
∣
∣
−AnA(n+1)
n− 1 ,n+1;rn=0, (B 2 )
∣
∣
∣
∣
A
(n)
nr A(n)
nnA(n+1)
n+1,rA
(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)
rnA
(n+1)
n− 1 ,n+1;rnA
(n)
n− 1 ,rA
(n)
n− 1 ,nA
(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 ,nA
(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;rnA
(n−1)
n− 1 ,r−A
(n+1)
n− 1 ,n+1;rnA
(n−1)
rr+A
(n+1)
n,n+1;rnA
(n)
r,n−1;rn]
(3.6.18)
=A
(n−1)
n− 1 ,r∣
∣
∣
∣
A
(n)
rr A(n)
rnA(n+1)
n+1,rA
(n+1)
n+1,n∣
∣
∣
∣
−A
(n−1)
rr∣
∣
∣
∣
∣
A
(n)
n− 1 ,r A(n)
n− 1 ,nA
(n+1)
n+1,rA
(n+1)
n+1,n∣
∣
∣
∣
∣
+A
(n)
r,n−1;rn∣
∣
∣
∣
A
(n)
nr A(n)
nnA(n+1)
n+1,rA
(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)
rrA(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)
rnA(n−1)
n− 1 ,r A(n)
n− 1 ,n