Page 83 - Determinants and Their Applications in Mathematical Physics

68 4. Particular Determinants

Proof. Let A n = |a ij | n and let E n+1 and F n+1 denote determinants
obtained by bordering A n in different ways:
1 1 1 1 ···

−x • a 12 a 13 ···

E n+1 = −x −a 12 • a 23 ···

−x −a 13 −a 23 • ···

··· ··· ··· ··· ···
n+1
and F n+1 is obtained by replacing the first column of E n+1 by the column

T
0 − 1 − 1 − 1 ··· .
n+1
Both A n and F n+1 are skew-symmetric. Then,
E n+1 = A n + xF n+1 .
Return to E n+1 and perform the column operations
C = C j − C 1 , 2 ≤ j ≤ n +1,

j
which reduces every element to zero except the first in the first row and
increases every other element in columns 2 to (n +1) by x. The result is
E n+1 = |a ij + x| n .
Hence, applying Theorems 4.9 and 4.10,

|a ij + x| 2n = A 2n + xF 2n+1
= A 2n ,

|a ij + x| 2n−1 = A 2n−1 + xF 2n
= xF 2n .
The theorem follows.
Corollary. The determinant
A = |a ij | 2n , where a ij + a ji =2x,

can be expressed as a skew-symmetric determinant of the same order.
Proof. The proof begins by expressing A in the form
x

a 12 a 13 a 14 ···
x
2x − a 12 a 23 a 24 ···

A = 2x − a 13 2x − a 23 x a 34 ···

2x − a 14 2x − a 24 2x − a 34 x ···

··· ··· ··· ··· ···
2n
and is completed by subtracting x from each element.
Let
A n = |a ij | n , a ji = −a ij ,