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

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

  3. 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 when we were finding a polynomial that passed through a given set of points.