Vandermonde matrix and determinant
I failed to mention the Vandermonde matrix because I couldn't see how it fit in with the story. Let
P(x)  = (x  x_{1})· ... ·(x  x_{n}) 
= x^{n} + a_{1}x^{n1} + a_{2}x^{n2} + ... + a_{n}. 
Let
G = 

be the companion matrix of P(x). Then
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