Page 92 - A Course in Linear Algebra with Applications

76 Chapter Three: Determinants

Let D be the value of the determinant. Clearly it is a
polynomial in xi, x 2,..., x n. If we apply the column operation
j
Ci—Cj , with i < , to the determinant, its value is unchanged.
On the other hand, after this operation each entry in column i
will be divisible by xj — Xj. Hence D is divisible by •X>£ Jb n for
2
alH, j ; = 1, ,..., n and i < . Thus we have located a total of
j
n(n— )/2 distinct linear polynomials which are factors of D,
l
this being the number of pairs of distinct positive integers i,j
such that 1 < i < j < n. But the degree of the polynomial D
is equal to

, „ N n(n — 1)
l + 2 + --- + ( n - l ) = 2 •

for each term in the denning sum has this degree. Hence D
must be the product of these n(n —1)/2 factors and a constant
c, there being no room for further factors. Thus

D = cY[{xi-Xj),

with i < j = 1, ,..., n. In fact c is equal to 1, as can be seen
2
by looking at the coefficient of the term lx 2x^ • • • x^~ l in the
defining sum for the determinant D; this corresponds to the
permutation 1,2,... ,n, and so its coefficient is +1. On the
other hand, in the product of the x^ — Xj the coefficient of the
term is 1. Hence c = 1.
The critical property of the Vandermonde determinant D
is that D = 0 if and only if at least two of xi, x 2,..., x n are
equal.

Exercises 3.2
1. By using elementary row operations compute the following
determinants:
1 0 3 2
1 4 2 3 1 - 2 3 4 - 1 2
- 2 4 7 , (b) 0 4 4 , (c) 0 3 1 2 '
6 1 2 2 - 3 6
1 5 2 3