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let-A-n-cos-npi-3-sin-npi-3-sin-npi-3-cos-npi-3-1-calculate-A-0-A-1-and-A-2-2-calculate-det-A-n-is-A-n-inversible-3-calculste-A-n-n-4




Question Number 130064 by mathmax by abdo last updated on 22/Jan/21
let A_n = (((cos(((nπ)/3))            sin(((nπ)/3)))),((sin(((nπ)/3))                 cos(((nπ)/3)))) )  1) calculate A_0 ,A_1  and A_2   2) calculate det(A_n ) is A_n inversible?  3) calculste A_n ^n   4) find lim_(n→+∞)  A_n ^n
$$\mathrm{let}\:\mathrm{A}_{\mathrm{n}} =\begin{pmatrix}{\mathrm{cos}\left(\frac{\mathrm{n}\pi}{\mathrm{3}}\right)\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{sin}\left(\frac{\mathrm{n}\pi}{\mathrm{3}}\right)}\\{\mathrm{sin}\left(\frac{\mathrm{n}\pi}{\mathrm{3}}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{cos}\left(\frac{\mathrm{n}\pi}{\mathrm{3}}\right)}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{A}_{\mathrm{0}} ,\mathrm{A}_{\mathrm{1}} \:\mathrm{and}\:\mathrm{A}_{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{calculate}\:\mathrm{det}\left(\mathrm{A}_{\mathrm{n}} \right)\:\mathrm{is}\:\mathrm{A}_{\mathrm{n}} \mathrm{inversible}? \\ $$$$\left.\mathrm{3}\right)\:\mathrm{calculste}\:\mathrm{A}_{\mathrm{n}} ^{\mathrm{n}} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{find}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{A}_{\mathrm{n}} ^{\mathrm{n}} \\ $$

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