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how to go about computing following determinant? I tried using Gaussian elimination on some special cases and figured there might be some pattern, maybe a recurrence relation involved, but I just can't see it.

$\begin{vmatrix} 1 & 2 & 3 & \cdots & n-1 & n \\ 2 & 3 & 4 & \cdots & n & 1 \\ 3 & 4 & 5 & \cdots & 1 & 2 \\ \vdots & \vdots & \ddots & \vdots \\ n & 1 & 2 & \cdots & n-2 & n-1 \end{vmatrix}$

user135482
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  • Here's a guess: multiply the ith row by n!/i. It'll make every entry in the first column identical. – Addem Mar 14 '14 at 18:01
  • This is a circulant matrix http://en.wikipedia.org/wiki/Circulant_matrix for which there is an explicit determinant formula. This may however be somehow difficult to calculate. – Lord Soth Mar 14 '14 at 19:41
  • @LordSoth Yes, it is easy to see what the eigenvalues are. But how to derive an explicit formula for their product? – mirgee Mar 14 '14 at 20:08

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