Determinants and Their Applications in Mathematical Physics

188 5. Further Determinant Theory

and

uij=

1

xi−xj

=−uji, (5.3.2)

where thexiare distinct but otherwise arbitrary.

Illustration.

A 3 =

∣ ∣ ∣ ∣ ∣ ∣

x−u 12 −u 13 u 12 u 13

u 21 x−u 21 −u 23 u 23

u 31 u 32 x−u 31 −u 32

∣ ∣ ∣ ∣ ∣ ∣

.

Theorem.

An=x

n

.

[This theorem appears in a section of Matsuno’s book in which thexi

are the zeros of classical polynomials but, as stated above, it is valid for all

xi, provided only that they are distinct.]

Proof. The sum of the elements in each row isx. Hence, after performing

the column operations

C

′

n

=

n

∑

j=1

Cj

=x

[

111 ··· 1

]

T

,

it is seen thatAnis equal toxtimes a determinant in which every element

in the last column is 1. Now, perform the row operations

R

′

i

=Ri−Rn, 1 ≤i≤n− 1 ,

which remove every element in the last column except the element 1 in

position (n, n). The result is

An=xBn− 1 ,

where

Bn− 1 =|bij|n− 1 ,

bij=











uij−unj=

uijunj

uni

,j=i

x−

n∑− 1

r=1

r=i

uir,j=i.

It is now found that, after rowihas been multiplied by the factoruni,

1 ≤i≤n−1, the same factor can be canceled from columni,1≤i≤n−1,

to give the result

Bn− 1 =An− 1.