Chapter 2.2, Problem 12E

Consider the $3\times 3$ Vandermonde matrix

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

(a) Show that $\det \left(V\right)=\left(x_2-x_1\right)\left(x_3-x_1\right)\left(x_3-x_2\right)$ . Hint: Make use of row operation III.

(b) What conditions must the scalars $x_1,x_2\text{and}{x}_3,$ satisfy in order for V to be nonsingular?