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Hi!

Perhaps it is an easy question but i don't figure out how to prove it. Let $a_1,...,a_{2m+2}\in\mathbb{C}$ and for $1\leq i\leq 2m+2$ and $j\leq [\frac{2m+2-i}{2}]$ (with $[a]$ i mean the integer part of $a$) consider the matrix $A(i,j)=(a_{i+k+l})_{0\leq k,l\leq j}$. I denote with $D(i,j)=\det(A(i,j))$. I have to prove the following relation:$$D(1,m-1)D(3,m-1)-D(1,m)D(3,m-2)=D(2,m-1)^2$$

I tried by induction but i didn't manage to get the result.

Any help or hint would be appreciated.

Thank you in advance.

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  • $\begingroup$ First : did you try in low dimension, to get a feel of the problem. Second : is it homework you want done? $\endgroup$
    – Julien Puydt
    Commented Feb 3, 2011 at 15:53
  • $\begingroup$ @Snark: yes i tried in low dimension and the result is correct, i didn't find this relation as an exercise or homework but in an article i'm reading. It is said "well known identity on determinants" so i tried to prove it because i didn't know that identity. $\endgroup$
    – Italo
    Commented Feb 3, 2011 at 17:39

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This must be a consequence od the Dodgson's condensation formula. See Exercise 24 on my web site. By the way, C. L. Dodgson was the real name of Lewis Caroll (see the MO question link text.

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