###### Exercise 13

The following matrix is called a *Vandermond* matrix:

\begin{equation*}
V = \left[\begin{array}{rrr}
1 \amp a \amp a^2 \\
1 \amp b \amp b^2 \\
1 \amp c \amp c^2 \\
\end{array}\right]\text{.}
\end{equation*}

Use row operations to explain why \(\det V = (b-a)(c-a)(c-b)\text{.}\)

Explain why \(V\) is invertible if \(a\text{,}\) \(b\text{,}\) and \(c\) are all distinct real numbers.

There is a natural way to generalize this to a \(4\times4\) matrix with parameters \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\text{.}\) Write this matrix and state its determinant based on your previous work.

This matrix appeared in Exercise 1.4.4.7 when we were finding a polynomial that passed through a given set of points.