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compute \begin{vmatrix} \lambda & 1 & & & & \\ n & \lambda & 2 & & & \\ & \ddots & \ddots & \ddots & & \\ & & \ddots & \ddots & \ddots & \\ & & & 2 & \lambda & n \\ & & & & 1 & \lambda \end{vmatrix}

I tried to find some recurrence relations, but I failed. I added all the rows except the first row to the first row, but I can't find any useful equations. How to compute it?

Jean Marie
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G. M.
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  • It is unclear what the pattern of the subdiagonal element is, could there be a typo? – Martin R Nov 27 '22 at 17:34
  • Oh you're right. There is a typo. – G. M. Nov 27 '22 at 17:37
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    You say you've tried some recurrence relations. Can we see what exactly you've tried, and how you derive these recurrence relations? – V.S.e.H. Nov 27 '22 at 17:41
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    please, show the first values, for $n=1,2,3,4$ – Daniel S. Nov 27 '22 at 17:47
  • Emmm...if the subdiagonal elements are the same, I can find some recurrence relations to get the result, since I've computed some example. Unfortunately, the subdiagonal elements are not equal, and i think that's the difficulty with calculating it. Now i might already know the answer after computing the determinant when n=3, 4, 5, 6, but I still want a general solution. – G. M. Nov 27 '22 at 18:01

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