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Question Number 205669 by hardmath last updated on 26/Mar/24

Answered by Berbere last updated on 27/Mar/24

$$A=X^5$$$${\chi }_A=(3-x)(5-x)-8=X^2-8X+7$$$$(x-1)(X-7)D=\left(\begin{array}{c}10\\ 07\end{array}\right)$$$$U_1=\left(\begin{array}{c}1\\ -1\end{array}\right),U_2\left(\begin{array}{c}1\\ 2\end{array}\right);P=\left(\begin{array}{c}11\\ -12\end{array}\right)$$$$P^{-1}=\frac{1}{3}\left(\begin{array}{c}2-1\\ 11\end{array}\right)$$$$A={PDP}^{-1};X={PD}^{\frac{1}{5}}{P}^{-1}=\frac{1}{3}.\left(\begin{array}{c}2-1\\ 11\end{array}\right)\left(\begin{array}{c}10\\ 0\sqrt[5]{7}\end{array}\right)\left(\begin{array}{c}2-1\\ 11\end{array}\right)$$$$=\frac{1}{3}\left(\begin{array}{c}4-\sqrt[5]{7}-2-\sqrt[5]{7}\\ 2+\sqrt[5]{7}-1+\sqrt[5]{7}\end{array}\right);\mathrm{\Omega }=3$$

Commented by hardmath last updated on 27/Mar/24

$$\mathrm{Thank}\mathrm{you}\mathrm{dear}\mathrm{professor}\mathrm{cool}$$

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