Chapter 6.2, Problem 31E

Vandermonde determinants (introduced by Alexandre-Théophile Vandermonde). Consider distinct real numbers $a_0,a_1,\mathrm{...},a_n$ . We define the $\left(n+1\right)\times \left(n+1\right)$ matrix
$A=\left[\begin{array}{cccc}1& 1& \cdots & 1\\ a_0& a_1& \cdots & a_n\\ a_0^2& a_1^2& \cdots & a_n^2\\ ⋮& ⋮& & ⋮\\ a_0^n& a_1^n& \cdots & a_n^n\end{array}\right]$ .

Vandermonde showed that $\det \left(A\right)={\displaystyle \underset{i>j}{\prod }\left(a_i-a_i\right)}$ ,
the product of all differences $\left(a_i-a_i\right)$ , where ¡ exceeds j.
a. Verify this formula in the case of $n=1$ .
b. Suppose the Vandermonde formula holds for $n-1$ .
You are asked to demonstrate it for n. Consider the function
$f\left(t\right)=\det \left[\begin{array}{ccccc}1& 1& \cdots & 1& 1\\ a_0& a_1& \cdots & a_{n-1}& t\\ a_0^2& a_1^2& \cdots & a_{n-1}^2& t^2\\ ⋮& ⋮& & ⋮& ⋮\\ a_0^n& a_1^n& \cdots & a_{n-1}^n& t^n\end{array}\right]$ .
Explain why $f\left(t\right)$ is a polynomial of nth degree. Find the coefficient k of $t^n$ using Vandermonde’s formula for $a_0,\mathrm{...},a_{n-1}$ . Explain why $f\left(a_0\right)=f\left(a_1\right)=\cdots =f\left(a_{n-1}\right)=0$ .
Conclude that $f\left(t\right)=k\left(t-a_0\right)\left(t-a_1\right)\cdots \left(t-a_{n-1}\right)$ for the scalar k you found above. Substitute $t=a_n$ to demonstrate Vandermonde’s formula.