Page 68 - Determinants and Their Applications in Mathematical Physics

4.1 Alternants 53

When any two of the x r are equal, X n has two identical rows and therefore
vanishes. Hence, very possible difference of the form (x s − x r ) is a factor
of X n , that is,
X n = K(x 2 − x 1 )(x 3 − x 1 )(x 4 − x 1 ) ··· (x n − x 1 )
(x 3 − x 2 )(x 4 − x 2 ) ··· (x n − x 2 )
(x 4 − x 3 ) ··· (x n − x 3 )
······
(x n − x n−1 )

= K (x s − x r ),
1≤r<s≤n
1
which is the product of K and n(n − 1) factors. One of the terms in the
2
expansion of this polynomial is the product of K and the first term in each
factor, namely
2 3
Kx 2 x x ··· x n−1 .
3 4
n
Comparing this term with (4.1.4), it is seen that K = 1 and the theorem
is proved.
Second Proof. Perform the column operations

C = C j − x n C j−1
j
in the order j = n, n−1,n−2,..., 3, 2. The result is a determinant in which
the only nonzero element in the last row is a 1 in position (n, 1). Hence,

X n =(−1) n−1 V n−1 ,

where V n−1 is a determinant of order (n − 1). The elements in row s of
V n−1 have a common factor (x s − x n ). When all such factors are removed
from V n−1 , the result is
n−1

(x n − x r ),
X n = X n−1
r=1
which is a reduction formula for X n . The proof is completed by reducing
the value of n by 1 repeatedly and noting that X 2 = x 2 − x 1 .

Exercises

1. Let

j − 1

A n = (−x i ) j−i =1.
i − 1

n