Determinants and Their Applications in Mathematical Physics

68 4. Particular Determinants

Proof. LetAn=|aij|nand letEn+1 andFn+1denote determinants

obtained by borderingAnin different ways:

En+1=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

11 11···

−x • a 12 a 13 ···

−x −a 12 • a 23 ···

−x −a 13 −a 23 • ···

··· ··· ··· ··· ···

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

andFn+1is obtained by replacing the first column ofEn+1by the column

[

0 − 1 − 1 − 1 ···

]T

n+1

.

BothAnandFn+1are skew-symmetric. Then,

En+1=An+xFn+1.

Return toEn+1and perform the column operations

C

′

j=Cj−C^1 ,^2 ≤j≤n+1,

which reduces every element to zero except the first in the first row and

increases every other element in columns 2 to (n+1) byx. The result is

En+1=|aij+x|n.

Hence, applying Theorems 4.9 and 4.10,

|aij+x| 2 n=A 2 n+xF 2 n+1

=A 2 n,

|aij+x| 2 n− 1 =A 2 n− 1 +xF 2 n

=xF 2 n.

The theorem follows.

Corollary. The determinant

A=|aij| 2 n, where aij+aji=2x,

can be expressed as a skew-symmetric determinant of the same order.

Proof. The proof begins by expressingAin the form

A=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

xa 12 a 13 a 14 ···

2 x−a 12 xa 23 a 24 ···

2 x−a 13 2 x−a 23 xa 34 ···

2 x−a 14 2 x−a 24 2 x−a 34 x ···

··· ··· ··· ··· ···

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2 n

and is completed by subtractingxfrom each element.

Let

An=|aij|n,aji=−aij,