5
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$$\left| {\begin{array}{*{20}{c}} 1&{{a_1} + a_1^{ - 1}}& \cdots &{a_1^{n - 1} + a_1^{n - 3} + a_1^{n - 5} + \cdots + a_1^{1 - n}}\\ 1&{{a_2} + a_2^{ - 1}}& \cdots &{a_2^{n - 1} + a_2^{n - 3} + a_2^{n - 5} + \cdots + a_2^{1 - n}}\\ \vdots & \vdots & \vdots & \vdots \\ 1&{{a_n} + a_n^{ - 1}}& \cdots &{a_n^{n - 1} + a_n^{n - 3} + a_n^{n - 5} + \cdots + a_n^{1 - n}} \end{array}} \right|$$

How to calculate it? Is it possible to transform it into Vandermonde?

I want to know whether it can never be zero if $a_i\ne a_j$. Any suggestion is appreciated!

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  • 1
    $\begingroup$ Notice for any $k > 0$, $a^k + a^{k-2} + \cdots + a^{2-k} + a^{-k} = P(a + a^{-1})$ for some monic polynomial $P(x)$ of degree $k$. So the determinant is equivalent to a Vandermonde determinant with entries $(a_i+a_i^{-1})^{j-1}$ $\endgroup$ Commented May 20, 2015 at 22:44
  • $\begingroup$ Get it! Thank you so much! $\endgroup$
    – Aborna
    Commented May 20, 2015 at 22:59

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