3.6 The Jacobi Identity and Variants 41
=
∣ ∣ ∣ ∣ ∣ ∣ ∣
Aa 21 a 23
Aa 31 a 33
a 11 a 13
a 41 a 43
∣ ∣ ∣ ∣ ∣ ∣ ∣
=A
2
∣
∣
∣
∣
a 11 a 13
a 41 a 43
∣
∣
∣
∣
=A
2
M 23 , 24
=σA
2
A 23 , 24.
Hence, transposingJ,
J=
∣
∣
∣
∣
A 22 A 24
A 32 A 34
∣
∣
∣
∣
=AA 23 , 24
which completes the illustration.
Restoring the parametern, the Jacobi identity withr =2,3 can be
expressed as follows:
r=2:
∣
∣
∣
∣
∣
A
(n)
ip
A
(n)
iq
A
(n)
jp
A
(n)
jq
∣
∣
∣
∣
∣
=AnA
(n)
ij,pq. (3.6.4)
r=3:
∣ ∣ ∣ ∣ ∣ ∣ ∣
A
(n)
ip A
(n)
iq A
(n)
ir
A
(n)
jp
A
(n)
jq
A
(n)
jr
A
(n)
kp
A
(n)
kq
A
(n)
kr
∣ ∣ ∣ ∣ ∣ ∣ ∣
=A
2
n
A
(n)
ijk,pqr
. (3.6.5)
3.6.2 The Jacobi Identity — 2
The Jacobi identity for small values ofrcan be proved neatly by a technique
involving partial derivatives with respect to the elements ofA. The general
result can then be proved by induction.
Theorem 3.4. For an arbitrary determinantAnof ordern,
∣
∣
∣
∣
A
ij
n
A
iq
n
A
pj
n
A
pq
n
∣
∣
∣
∣
=A
ip,jq
n
,
where the cofactors are scaled.
Proof. The technique is to evaluate∂A
ij
/∂apqby two different methods
and to equate the results. From (3.2.15),
∂A
ij
∂apq
=
1
A
2
[
AAip,jq−AijApq
]
. (3.6.6)
Applying double-sum identity (B) in Section 3.4,
∂A
ij
∂apq
=−
∑
r
∑
s
∂ars
∂apq
A
is
A
rj