Page 203 - Determinants and Their Applications in Mathematical Physics

188 5. Further Determinant Theory

and
1
u ij = x i − x j = −u ji , (5.3.2)
where the x i are distinct but otherwise arbitrary.

Illustration.

x − u 12 − u 13 u 12 u 13

A 3 = u 21 x − u 21 − u 23 u 23 .

u 31 u 32 x − u 31 − u 32

Theorem.
A n = x .
n
[This theorem appears in a section of Matsuno’s book in which the x i
are the zeros of classical polynomials but, as stated above, it is valid for all
x i , provided only that they are distinct.]
Proof. The sum of the elements in each row is x. Hence, after performing
the column operations
n

C =
n C j
j=1
T
= x 111 ··· 1 ,
it is seen that A n is equal to x times a determinant in which every element
in the last column is 1. Now, perform the row operations

R = R i − R n , 1 ≤ i ≤ n − 1,
i
which remove every element in the last column except the element 1 in
position (n, n). The result is
A n = xB n−1 ,
where

B n−1 = |b ij | n−1 ,
u ij − u nj = ,j = i
 u ij u nj
 u ni

n−1
b ij = !
x − u ir , j = i.
r=1


r =i
It is now found that, after row i has been multiplied by the factor u ni ,
1 ≤ i ≤ n−1, the same factor can be canceled from column i,1 ≤ i ≤ n−1,
to give the result
B n−1 = A n−1 .