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 ... 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