Exercise 13

The following matrix is called a Vandermond matrix: $$V = \left[\begin{array}{rrr} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \\ \end{array}\right]$$.

1. Use row operations to explain why $\det(V) = (b-a)(c-a)(c-b)$.

2. Explain why $V$ is invertible if and only if $a$, $b$, and $c$ are all distinct real numbers.

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

This matrix appeared in this exercise when we were found a polynomial that passed through a given set of points.