Find the determinant of a Vandermonde-like matrix

Question

I reduced a certain determinant to the following which has the form: \begin{vmatrix} 1&1&\cdots & 1 \\ -a_1&a_2&\cdots &a_n\\ -a_1^{2}&a_2^{2}&\cdots&a_n^{2}\\ \vdots&\vdots&\ddots& \vdots\\ -a_1^{n-1}&a_2^{n-1}&\cdots&a_n^{n-1}\\ \end{vmatrix}

To clarify a bit, it is exactly the Vandermonde determinant except the first column is negative, while the $(1,1)$-entry is still $1$.

I think the key is to apply Vandermonde determinant yet I can't proceed. There may be a quick answer to this, however. Any hints?

Answer 1

By linearity in or Laplace expansion along the first column, the determinant is equal to $$ \begin{vmatrix} -1&1&\cdots & 1 \\ -a_1&a_2&\cdots &a_n\\ -a_1^{2}&a_2^{2}&\cdots&a_n^{2}\\ \vdots&\vdots&\ddots& \vdots\\ -a_1^{n-1}&a_2^{n-1}&\cdots&a_n^{n-1}\\ \end{vmatrix} + 2\begin{vmatrix} a_2&\cdots &a_n\\ a_2^{2}&\cdots&a_n^{2}\\ \vdots&\ddots& \vdots\\ a_2^{n-1}&\cdots&a_n^{n-1}\\ \end{vmatrix}. $$ Now each of the two summands is a multiple of a Vandermonde determinant.

Answer 2

It is also possible to evaluate a more general determinant, where the clever observation of @user1551 isn't available. Let $c_0,\dots, c_1$ be arbitrary and let $X$ be a polynomial variable. Then we will evaluate $$ \begin{vmatrix} c_0&1&\cdots & 1 \\ c_1 X&a_2&\cdots &a_n\\ c_2 X^2&a_2^{2}&\cdots&a_n^{2}\\ \vdots&\vdots&\ddots& \vdots\\ c_{n-1}X^{n-1}&a_2^{n-1}&\cdots&a_n^{n-1}\\ \end{vmatrix}. $$

By a Lagrange expansion on the first column we get that this is $$ \sum_{0}^{n-1} (-1)^{k} c_k X^k V_k \hspace{10em}\text{(*)} $$ where $V_k$ is the determinant of the corresponding $(n-1)\times(n-1)$ minor in $$ \begin{vmatrix} 1&\cdots & 1 \\ a_2&\cdots &a_n\\ a_2^{2}&\cdots&a_n^{2}\\ \vdots&\ddots& \vdots\\ a_2^{n-1}&\cdots&a_n^{n-1}\\ \end{vmatrix}. $$ But now consider the special case where all $c_k=1$; this is just the standard Vandermonde determinant, so we have $$ \sum_{0}^{n-1} (-1)^{k} X^k V_k = \prod_{r=2}^{n} (X-a_r)\prod_{2\leqslant s < t \leqslant n} (a_t - a_s). $$

We write, in the usual way, $$ \prod_{r=2}^{n} (X-a_r)=\sum_{k=0}^{n-1}(-1)^{k} \sigma_{n-1-k} X^k; $$ these are just the usual symmetric functions of the $a_2,\dots, a_{n}$ taking their sum, their sum two at a time, and so on.

The coefficients $V_k$ then all have a common factor $\prod_{2\leqslant s < t \leqslant n} (a_t - a_s)$, the Vandermonde of $a_2, \dots, a_n$ (which deals with the skew-symmetry in these variables); the other factor in each case is just the symmetric function $\sigma_{n-k}$: $$ V_k= \sigma_{n-k}\prod_{2\leqslant s < t \leqslant n} (a_t - a_s).$$

Taking $c_0=1, c_1=\dots=c_n=1$ and substituting $X=a_1$ in (*) gives the value for the determinant of the original question.