# Find mod ( det A )

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• Dec 17th 2011, 01:19 AM
FernandoRevilla
Find mod ( det A )
Challenge problem

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

$\displaystyle A=\begin{bmatrix} 1 & 1 & 1 & \ldots & 1\\ 1 & \epsilon & \epsilon^2 & \ldots & \epsilon^{n-1} \\ 1 & \epsilon^2 & \epsilon^4 & \ldots & \epsilon^{2n-2}\\ \vdots&&&&\vdots \\ 1 & \epsilon^{n-1} & \epsilon^{2(n-1)}&\ldots & \epsilon^{(n-1)^2}\end{bmatrix}\quad (\epsilon=\cos(2\pi/n)+i\sin (2\pi/n))$
• Dec 17th 2011, 06:04 PM
NonCommAlg
Re: Find mod ( det A )
Quote:

Originally Posted by FernandoRevilla
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?
• Dec 17th 2011, 11:06 PM
FernandoRevilla
Re: Find mod ( det A )
Quote:

Originally Posted by NonCommAlg
what does$\displaystyle \mod (\det A)$ mean?

Modulus of the complex number $\displaystyle \det (A)$ .

Quote:

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

Quote:

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 $\displaystyle \mod (\det A)=n^{n/2}$ .
• Dec 18th 2011, 01:17 AM
NonCommAlg
Re: Find mod ( det A )
Quote:

Originally Posted by FernandoRevilla
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 $\displaystyle a.$
• Dec 18th 2011, 04:28 AM
FernandoRevilla
Re: Find mod ( det A )
A better approach is to find previously $\displaystyle A^2$ .