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Thread: Find mod ( det A )

  1. #1
    MHF Contributor FernandoRevilla's Avatar
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    Find mod ( det A )

    Challenge problem

    Find $\displaystyle \mod (\det A)$ being $\displaystyle A\in\textrm{Mat}_{n\times n}(\mathbb{C})$ defined by

    $\displaystyle A=\begin{bmatrix} 1 & 1 & 1 & \ldots & 1\\ 1 & \epsilon & \epsilon^2 & \ldots & \epsilon^{n-1} \\ 1 & \epsilon^2 & \epsilon^4 & \ldots & \epsilon^{2n-2}\\ \vdots&&&&\vdots \\ 1 & \epsilon^{n-1} & \epsilon^{2(n-1)}&\ldots & \epsilon^{(n-1)^2}\end{bmatrix}\quad (\epsilon=\cos(2\pi/n)+i\sin (2\pi/n))$
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    Re: Find mod ( det A )

    Quote Originally Posted by FernandoRevilla View Post
    Challenge problem

    Find $\displaystyle \mod (\det A)$ being $\displaystyle A\in\textrm{Mat}_{n\times n}(\mathbb{C})$ defined by

    $\displaystyle A=\begin{bmatrix} 1 & 1 & 1 & \ldots & 1\\ 1 & \epsilon & \epsilon^2 & \ldots & \epsilon^{n-1} \\ 1 & \epsilon^2 & \epsilon^4 & \ldots & \epsilon^{2n-2}\\ \vdots&&&&\vdots \\ 1 & \epsilon^{n-1} & \epsilon^{2(n-1)}&\ldots & \epsilon^{(n-1)^2}\end{bmatrix}\quad (\epsilon=\cos(2\pi/n)+i\sin (2\pi/n))$
    what does$\displaystyle \mod (\det A)$ mean? obviously $\displaystyle \det A=\prod_{r < s}(\epsilon^s - \epsilon^r).$ maybe you want a more simplified answer?
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Re: Find mod ( det A )

    Quote Originally Posted by NonCommAlg View Post
    what does$\displaystyle \mod (\det A)$ mean?
    Modulus of the complex number $\displaystyle \det (A)$ .

    obviously $\displaystyle \det A=\prod_{r < s}(\epsilon^s - \epsilon^r).$
    Right, $\displaystyle A$ is a Vandermonde matrix.

    maybe you want a more simplified answer?
    Being this the Math Challenge Problem forum, I know the solution (and its resolution) . Another version is: prove that $\displaystyle \mod (\det A)=n^{n/2}$ .
    Last edited by FernandoRevilla; Dec 18th 2011 at 12:06 AM.
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    Re: Find mod ( det A )

    Quote Originally Posted by FernandoRevilla View Post
    Modulus of the complex number $\displaystyle \det (A)$ .

    Right, $\displaystyle A$ is a Vandermonde matrix.

    Being this the Math Challenge Problem forum, I know the solution (and its resolution) . Another version is: prove that $\displaystyle \mod (\det A)=n^{n/2}$ .
    so the question is to find $\displaystyle |\det(A)|.$ so far i got $\displaystyle |\det(A)| = 2^{\binom{n}{2}}a^n,$ where $\displaystyle a = \prod_{k=1}^m \sin(k \theta)$ and $\displaystyle \theta = \frac{\pi}{n}$ and $\displaystyle m = \lfloor \frac{n-1}{2} \rfloor.$

    so the problem now is to evaluate $\displaystyle a.$
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Find mod ( det A )

    A better approach is to find previously $\displaystyle A^2$ .
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