Question Number 50386 by Abdo msup. last updated on 16/Dec/18
$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{0}\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:−\mathrm{2}\:\:−\mathrm{9}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{4}\:\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\left({A}−{I}\right)^{\mathrm{3}} \\ $$$$\left.\mathrm{2}\right)\:{conclude}\:\:{A}^{{n}} \:{for}\:{n}\:\:{integr}. \\ $$
Answered by kaivan.ahmadi last updated on 16/Dec/18
$$\mathrm{A}−\mathrm{I}=\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:−\mathrm{3}\:\:\:−\mathrm{9}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{3}\right) \\ $$$$\left(\mathrm{A}−\mathrm{I}\right)^{\mathrm{2}} =\begin{pmatrix}{\mathrm{0}\:\:\:\:\mathrm{0}\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\mathrm{0}\:\:\:\:\:\mathrm{0}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\mathrm{0}\right) \\ $$$$\Rightarrow\left(\mathrm{A}−\mathrm{I}\right)^{\mathrm{3}} =\mathrm{0} \\ $$