mathematica.gr

Δημοσιεύτηκε: **Τετ Μάιος 20, 2020 8:42 am**

$A= \begin{pmatrix} 1 & 2 & 3 & \cdots & n\\ n & 1 & 2 & \cdots & n - 1\\ n - 1 & n & 1 & \cdots & n - 2\\ \cdots & \cdots & \cdots & \cdots & \cdots\\2 & 3 & 4 & \cdots & 1 \end{pmatrix}$

Δείξτε ότι
$\displaystyle{\epsilon^ndet\left(I_n-A^{2n}\right)+\epsilon^{n-1}det\left(\epsilon I_n-A^{2n}\right)+\epsilon^{n-2}det\left(\epsilon^2 I_n-A^{2n}\right)+\cdots +det\left(\epsilon^n I_n-A^{2n}\right)}$ $\displaystyle{=n(-1)^{n-1}\left[\frac{n^n(n+1)}{2}\right]^{2n^2-4n}\left(1+(n+1)^{2n}\left(2n+(-1)^n{{2n}\choose{n}}\right)\right)}$
όπου $\epsilon \in \mathbb{C}\backslash \mathbb{R}$
$\epsilon^{n+1}=1$

Δημοσιεύτηκε: **Πέμ Απρ 15, 2021 10:24 pm**

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mathematica.gr

2019 Jozsef Wildt International Math Competition (44)

2019 Jozsef Wildt International Math Competition (44)

Re: 2019 Jozsef Wildt International Math Competition (44)