Page 54 - Determinants and Their Applications in Mathematical Physics

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 ... A r1 a r+1,r+1 a r+1,r+2 ... a r+1,n

A 12 A 22 ... A r2 r−1 a r+2,r+1 a r+2,r+2 ... a r+2,n
= A .

.................... ...............................
... a n,r+1 a n,r+2 ...

A 2r
a nn
A 1r
A rr r
n−r
(3.6.2)
Proof. Raise the order of J 12...r;12...r from r to n by applying the Laplace
expansion formula in reverse as follows:
...
A 11 A r1
. .
. .
. .
r rows
...

A 1r A rr
. (3.6.3)

J 12...r;12...r = ...............................

A 1,r+1 ... A r,r+1 1
. . . (n − r)rows
. .
. . . .

... 1

A 1n
A rn
n
Multiply the left-hand side by A, the right-hand side by |a ij | n , apply the
formula for the product of two determinants, the sum formula for elements
and cofactors, and, finally, the Laplace expansion formula again
.

.
A . a 1,r+1 ...
a 1n
. . . .
. . . .
. . . .
.
r rows
.
A . ...
a r,r+1 a rn

AJ 12...r;12...r = .....................................
.
.
. a r+1,r+1
... a r+1,n (n − r)rows
. . .

. . . . . .

.
.
. a n,r+1 ... a nn

a r+1,r+1 ... a r+1,n

. .
. .
r
= A . .

a n,r+1 a nn
...
n−r
= A A 12...r;12...r .
r
The first stage of the proof follows.
The second stage proceeds as follows. Interchange pairs of rows and then
pairs of columns of adj A until the elements of J as defined in (3.6.1) appear