Chapter 2.1, Problem 43E

A matrix in which the entries in each row (or in each column) form a geometric progression starting with 1 is called a Vandermonde matrix in honor of the French medical doctor, mathematician, and musician AlexandreThéophile Vandermonde (February 28, 1735–January 1, 1796). Here are two examples.

$V=\left[\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right]andV=\left[\begin{array}{cccc}1& a& a^2& a^3\\ 1& b& b^2& b^3\\ 1& c& c^2& c^3\\ 1& d& d^2& d^3\end{array}\right]$

Vandermonde matrices arise in a variety of applications, such as polynomial interpolation (see Formula (14) and Example 6 of Section 1.10). Use cofactor expansion to prove that

$\left[\begin{array}{ccc}1& x_1& x_1^2\\ 1& x_2& x_2^2\\ 1& x_3& x_3^2\end{array}\right]=(x_2-x_1)(x_3-x_1)(x_3-x_2)$