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I want to know sign of determinant of the following matrix: $$\begin{vmatrix} x_1&x'_1&0&0&0&\cdots&0\\ x'_1&x_2&x'_2&0&0&\cdots&0\\ 0&x'_2&x_3&x'_3&0&\cdots&0\\ 0&0&x'_3&x_4&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\cdots&\vdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&0&0&\vdots&x_n\\ \end{vmatrix}$$ where $$x_i=\frac{2}{a^2(\frac{b}{a})^{(2i-1)/(n+1)}}, x_i'=-\frac{1}{a^2(\frac{b}{a})^{(2i)/(n+1)}},a>0, b>0$$ I know about recurrence relation for tridiagonal matrix, but it still isn't enough. Looks like answer depends on $a$, $b$ and $n$, but I need to know it anyway.

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