Page 84 - Determinants and Their Applications in Mathematical Physics

4.3 Skew-Symmetric Determinants 69

and let B n+1 denote the skew-symmetric determinant obtained by
bordering A n by the row

−1 − 1 − 1 ··· − 10
n+1
below and by the column
T
111 ··· 10
n+1
on the right.

Theorem 4.12 (Muir and Metzler). B n+1 is expressible as a skew-
symmetric determinant of order (n − 1).

Proof. The row and column operations

R = R i + a in R n+1 , 1 ≤ i ≤ n − 1,
i

C = C j + a jn C n+1 , 1 ≤ j ≤ n − 1,
j
when performed on B n+1 , result in the elements a ij and a ji being
∗
transformed into a ∗ and a , where
ij ji
∗
a = a ij − a in + a jn , 1 ≤ i ≤ n − 1,
ij
a = a ji − a jn + a in , 1 ≤ j ≤ n − 1,
∗
ji
∗
= −a .
ij
In particular, a ∗ = 0, so that all the elements except the last in both
in
column n and row n are reduced to zero. Hence, when a Laplace expansion
from the last two rows or columns is performed, only one term survives and
the formula
∗
B n+1 = |a | n−1
ij
emerges, which proves the theorem. When n is even, both sides of this
formula are identically zero.

4.3.2 Preparatory Lemmas

Let

where B n = |b ij | n
1, i<j − 1

b ij = 0, i = j − 1
−1, i>j − 1.