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9.4 VANDERMONDE AND RELATED DETERMINANTS

Determinants in which the elements of each column (or row) are the terms 1, r, r², ..., rⁿ^-1 of a geometric progression are called Vandermonde determinants, named for Alexandre-Théophile Vandermonde (1735-1796), who was the first to give a systematic treatment of the theory of determinants.

Let us evaluate the 4 by 4 Vandermonde determinant

The expansion by minors of first column entries as outlined in Problem 7, Section 9.3, yields the following:

Letting r, s, t, and u stand for the 3 by 3 determinants in this expression, we may write

D = f(x) = r - sx + tx² - ux³.
If we let x = a in D, two columns become identical and, by Problem 4, Section 9.3, D becomes zero. This means that f(a) = 0, and it follows from the Factor Theorem that x - a is a factor of f(x). Similarly, x - b and x - c are factors of f(x).

If two of the numbers a, b, and c are equal, D has identical columns and thus is zero. We therefore assume that a, b, and c are distinct numbers. Then f(x) is a multiple of the product of
x - a, x - b, and x - c. Since f(x) is a polynomial of degree 3 or less and has -u as the coefficient of x³, this means that f(x) must be -u(x - a)(x - b)(x - c). This can be written as
u(a - x)(b - x)(c - x).

We leave it as an exercise for the reader (in Problem 1 below) to show that the 3 by 3 determinant u is expressible as (b - a)(c - a)(c - b). Substituting this for u in the above gives the result:

D = (a - x)(b - x)(c - x)(b - a)(c - a)(c - b).

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Wednesday, June 10, 1998