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