TITLE: Calculate determinant of Vandermonde using specified steps.
QUESTION [5 upvotes]: $V_n(a_1,a_2\dots, a_n)$ is a $n\times n$ Vandermonde matrix = 
 $$\left|\begin{array}[cccc] 
11&a_1&\cdots&a^{n-1}_1\\
1&a_2&\cdots&a^{n-1}_2\\
1&a_3&\cdots&a^{n-1}_3\\
\vdots&\vdots&\ddots&\vdots\\
1&a_n&\cdots&a_n^{n-1}
\end{array}\right|$$
Replace $a_1$ by $x$ so that the first row is $1,x, \dots ,x^{n-1}$
$$V_n(x,a_2\dots, a_n) = \left|\begin{array}[cccc] 
11&x&\cdots&x^{n-1}\\
1&a_2&\cdots&a^{n-1}_2\\
1&a_3&\cdots&a^{n-1}_3\\
\vdots&\vdots&\ddots&\vdots\\
1&a_n&\cdots&a_n^{n-1}
\end{array}\right|$$
Let $P(x) = V_n(x,a_2\dots, a_n)$.
(a) Show that $P(x)$ is a polynomial in $x$ of degree $\leq n-1$. 
$z = 1\det() -x\det() + x^2\det() -\cdots+ (-1)^{n-1}x^{n-1} \cdot \det()$, where each $\det()$ stands for the determinant of a smaller matrix after removing the appropriate columns, rows. 
Is this right? But isn't $z$ a polynomial of degree $n$?
(b) Show that $P(x)$ has $n-1$ distinct roots $a_2, \dots a_n$ and therefore has factorization $P(x) = k \prod_{i=2}^n(x-a_i)$ where the constant factor $k$ is the coefficient of $x^{n-1}.$
I'll be honest, I have no clue how to do this. Not even sure where to start. Nor do I know where to start on the rest.
(c) Show that $k = (-1)^{n-1}V_{n-1}(a_2,\dots a_n)$.
(d) Use parts (b) and (c) to deduce the recursion formula $$V_n(a_1, \dots a_n) = \left(\prod_{i=2}^n(a_i-a_1)\right)V_{n-1}(a_2,\dots a_n)$$
(e) Use part (d) to deduce that $V_n(a_1,a_2,\dots a_n) = \prod^n_{1\leq i< j\leq n} (a_j - a_i)$

REPLY [2 votes]: (a) Show that $P(x)$ is a polynomial in $x$ of degree $\le n−1$.

Use the Leibniz formula for determinants, and show that each sum contains only one factor of $x^k$ from the top row.

(b) Show that $P(x)$ has $n−1$ distinct roots $a_2,~ \dots ,~ a_n$ and therefore has factorization $P(x)=k\prod^n_{i=2}(x−a_i)$ where the constant factor $k$ is the coefficient of $x^{n−1}.

Plug $x = a_i$ into your matrix, and compare the first row to the $i^{th}$ row.  Show that the determinant must be zero in that case.

(c) Show that $k = (-1)^{n-1}V_{n-1}(a_2,\dots a_n)$

This follows immediately from the Laplace expansion that you did in part (a).

(d) Use parts (b) and (c) to deduce the recursion formula...

Just plug $x=a_1$ and part (c) into the formula in (b).

(e) Use part (d) to deduce that...

Use the fact that $$\bigg(\prod_{x \in X} f_x\bigg)\bigg(\prod_{y \in Y} g_y\bigg) = \prod_{(x,y) \in X \times Y} f_x \cdot g_y$$