3.6 The Jacobi Identity and Variants 39
It is required to prove that
J 12 ...r;12...r=A
r− 1
M 12 ...r;12...r
=A
r− 1
A 12 ...r;12...r.
The replacement of the minor by its corresponding cofactor is permitted
since the sum of the parameters is even. In some detail, the simplified
theorem states that
∣ ∣ ∣ ∣ ∣ ∣ ∣
A 11 A 21 ... Ar 1
A 12 A 22 ... Ar 2
....................
A 1 r A 2 r ... Arr
∣ ∣ ∣ ∣ ∣ ∣ ∣ r
=A
r− 1
∣ ∣ ∣ ∣ ∣ ∣ ∣
ar+1,r+1 ar+1,r+2 ... ar+1,n
ar+2,r+1 ar+2,r+2 ... ar+2,n
...............................
an,r+1 an,r+2 ... ann
∣ ∣ ∣ ∣ ∣ ∣ ∣
n−r
.
(3.6.2)
Proof. Raise the order ofJ 12 ...r;12...rfromrtonby applying the Laplace
expansion formula in reverse as follows:
J 12 ...r;12...r=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∣ ∣ ∣ ∣ ∣ ∣
A 11 ... Ar 1
.
.
.
.
.
.
A 1 r ... Arr
...............................
A 1 ,r+1 ... Ar,r+1 1
.
.
.
.
.
.
.
.
.
A 1 n ... Arn 1
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∣ ∣ ∣ ∣ ∣ ∣ n
}
rrows
}
(n−r)rows
. (3.6.3)
Multiply the left-hand side byA, the right-hand side by|aij|n, apply the
formula for the product of two determinants, the sum formula for elements
and cofactors, and, finally, the Laplace expansion formula again
AJ 12 ...r;12...r=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
A
.
.
. a 1 ,r+1 ... a 1 n
.
.
.
.
.
.
.
.
.
.
.
.
A
.
.
. ar,r+1 ... arn
.....................................
.
.
. ar+1,r+1 ... ar+1,n
.
.
.
.
.
.
.
.
.
.
.
. an,r+1 ... ann
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
}
rrows
}
(n−r)rows
=A
r
∣ ∣ ∣ ∣ ∣ ∣ ∣
ar+1,r+1 ... ar+1,n
.
.
.
.
.
.
an,r+1 ... ann
∣ ∣ ∣ ∣ ∣ ∣ ∣
n−r
=A
r
A 12 ...r;12...r.
The first stage of the proof follows.
The second stage proceeds as follows. Interchange pairs of rows and then
pairs of columns of adjAuntil the elements ofJas defined in (3.6.1) appear