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I want to prove that $$\begin{align} \begin{vmatrix} a_1 & a_2 & ... a_{N+1} \\ 1 & 1 & ... 1 \\ a^{-1}_1 & a^{-1}_2 & ... a^{-1}_{N+1} \\ ... & ... & ... \\ a^{-N+1}_1 & a^{-N+1}_2 & ... a^{-N+1}_{N+1} \\ \end{vmatrix} &= \frac{\prod_{j=2}^{N+1}(a_1-a_j)}{a^{N-1}_1} \begin{vmatrix} 1 & ...& 1 \\ a^{-1}_2 & ...& a^{-1}_{N+1} \\ ... & ... & ... \\ a^{-N+1}_2 & ... &a^{-N+1}_{N+1} \\ \end{vmatrix} \\[2mm] &\\ & & \end{align}$$ I check that this relation holds for $N\in\{1,2\}$.Please help me find general proof.

Darek
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    If you multiply column $k$ of the left matrix by $a_k^{N-1}$, don't you get a vandermonde determinant? Similarly for the right matrix? – Gerry Myerson Oct 28 '22 at 09:00
  • Thanks. Yes, you are right. This is indeed a Vandermonde determinant. – Darek Oct 28 '22 at 09:21
  • I nearly see that if I multiply both sides by the product of the coefficients I obtain the Vandermonde determinant of size $N+1$ on the left hand side and of size $N$ on the right hand side? – Darek Oct 28 '22 at 09:25
  • Yeah, more or less. – Gerry Myerson Oct 28 '22 at 09:33
  • For this problem: normalize both matrices to Vandermonde and go to Wikipedia; keeping care to put the normalization back in:) Are you interested in the paper; that really twists the mind further in (as I recall) :) If so, I will try to unearth it from years of papers (composting digital math papers here). – rrogers Oct 28 '22 at 12:45
  • @rrogers it will be great :) – Darek Oct 29 '22 at 06:41
  • Here is what I found on my computer. Doesn't match what I remember, but don't be put of by the title the internal matter, page 5-... , deals directly and indirectly with VanderMonde matrices. https://arxiv.org/pdf/math/9902004.pdf The author has another later paper that deals with combinatorial matrices (i.e. pascal). Something I used extensively. Another fairly elementary reference: https://web.deu.edu.tr/halil.oruc/vandermondefactorization.pdf and a historical paper: https://arxiv.org/pdf/1204.4716.pdf Of course google scholar has many available varients. – rrogers Oct 30 '22 at 02:13

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