##### Exercise13

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

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