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I'm wondering how to find the value of this determinant.

$$\left[ {\begin{array}{*{20}{c}} 0&{{x_1}}&{{x_2}}&{{x_3}}& \ddots &{{x_n}} \\ {{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}}& \ddots \\ {{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}}&{{x_3}} \\ {{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}}&{{x_2}} \\ \ddots &{{x_3}}&{{x_2}}&{{x_1}}&0&{{x_1}} \\ {{x_n}}& \ddots &{{x_3}}&{{x_2}}&{{x_1}}&0 \end{array}} \right]$$

I tried for some small $n$ and couldn't see any clue to find its value expressed by an easy formula. Any help will be appreciated.

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  • $\begingroup$ Is this matrix $6 \times 6$ ? $\ \ \ (n=5)?$ $\endgroup$ – Widawensen Nov 16 '18 at 9:49
  • $\begingroup$ Anyway, at Wolphram Alpha operation det[{{0,x_1,x_2,x_3,x_4,x_5},{x_1, 0,x_1,x_2,x_3,x_4 },{x_2, x_1, 0,x_1,x_2,x_3 }, {x_3,x_2, x_1, 0,x_1,x_2 },{x_4, x_3,x_2, x_1, 0,x_1 },{x_5,x_4, x_3,x_2, x_1, 0 }}] doesn't provide easy to interpretation result. $\endgroup$ – Widawensen Nov 16 '18 at 9:59
  • $\begingroup$ see yutsumura.com/determinant-of-a-general-circulant-matrix $\endgroup$ – Aleksas Domarkas Nov 16 '18 at 10:29
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    $\begingroup$ This is a symmetric Toeplitz matrix. I am unaware of any closed formula for the determinant, but knowing your matrix has a name can possibly be of some help. $\endgroup$ – jobe Nov 16 '18 at 10:33
  • $\begingroup$ @Widawensen you are right $\endgroup$ – Aleksas Domarkas Nov 16 '18 at 10:43

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