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Math Help - Determinant with Exponents

  1. #1
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    Determinant with Exponents

    I was working on a problem, I need to show that a certain type of equation has a unique solution. By Cramer's Rule the problem reduces to:
    \left| \begin{array}{ccccc} 1^0&2^0&3^0&....&n^0\\ 1^1&2^1&3^1&....&n^1\\1^2&2^2&3^2&....&n^2\\....&.  ...&....&....&....\\1^n&2^n&3^n&....&n^n \end{array} \right| \not = 0
    I asked my teacher he said there is a name for it, but did not know much about it. Someone know what this is? Or how to prove it?
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    Thank you really helps. Now I can complete the proof.
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    You should have caught my mistake, it must have been,
    <br />
\left| \begin{array}{ccccc} 1^0&2^0&3^0&....&n^0\\ 1^1&2^1&3^1&....&n^1\\1^2&2^2&3^2&....&n^2\\....&.  ...&....&....&....\\1^{n-1}&2^{n-1}&3^{n-1}&....&n^{n-1} \end{array} \right| \not = 0<br />
    Because otherwise it is non-square.
    ---------
    Can you demonstrate that for the regular Vandermonde Matrix
    \Delta =\prod_{i>j}(x_i-x_j)
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  5. #5
    TD!
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    A proof can be found here.
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