Determinants and Their Applications in Mathematical Physics

4.3 Skew-Symmetric Determinants 69

and let Bn+1 denote the skew-symmetric determinant obtained by

borderingAnby the row

[

− 1 − 1 − 1 ···− 10

]

n+1

below and by the column

[

111 ··· 10

]T

n+1

on the right.

Theorem 4.12 (Muir and Metzler). Bn+1 is expressible as a skew-

symmetric determinant of order(n−1).

Proof. The row and column operations

R

′

i

=Ri+ainRn+1, 1 ≤i≤n− 1 ,

C

′

j

=Cj+ajnCn+1, 1 ≤j≤n− 1 ,

when performed on Bn+1, result in the elements aij and aji being

transformed intoa

∗

ijanda

∗

ji, where

a

∗

ij=aij−ain+ajn,^1 ≤i≤n−^1 ,

a

∗

ji

=aji−ajn+ain, 1 ≤j≤n− 1 ,

=−a

∗

ij

In particular,a

∗

in= 0, so that all the elements except the last in both

columnnand rownare reduced to zero. Hence, when a Laplace expansion

from the last two rows or columns is performed, only one term survives and

the formula

Bn+1=|a

∗

ij

|n− 1

emerges, which proves the theorem. Whennis even, both sides of this

formula are identically zero.

4.3.2 Preparatory Lemmas

Let

Bn=|bij|n

where

bij=

{

1 , i<j− 1

0 ,i=j− 1

− 1 , i>j−1.