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Question Number 196066 by sniper237 last updated on 17/Aug/23

$${\left(\begin{array}{c}1-1\\ -12\end{array}\right)}^{-8}=?$$

Answered by witcher3 last updated on 17/Aug/23

$$\left(\begin{array}{c}1-1\\ -12\end{array}\right)=\mathrm{A}$$$${\mathrm{A}}^2=\left(\begin{array}{c}2-3\\ -33\end{array}\right)=3\mathrm{A}-{\mathrm{I}}_2$$$$\Rightarrow \mathrm{A}(\mathrm{A}-{3\mathrm{I}}_2)=(\mathrm{A}-{3\mathrm{I}}_2)\mathrm{A}={\mathrm{I}}_2$$$${\mathrm{A}}^{-}=\mathrm{A}-{3\mathrm{I}}_2$$$${\mathrm{A}}^{-2}={\mathrm{A}}^2+{9\mathrm{I}}_2-6\mathrm{A}$$$$=-3\mathrm{A}+{8\mathrm{I}}_2$$$${\mathrm{A}}^{-4}={9\mathrm{A}}^2-48\mathrm{A}+{64\mathrm{I}}_2$$$$=-21\mathrm{A}+{55\mathrm{I}}_2$$$${\mathrm{A}}^{-8}=({441\mathrm{A}}^2+{3025\mathrm{I}}_2-2310\mathrm{A})$$$$=441(3\mathrm{A}-\mathrm{I})+3025\mathrm{I}-2310\mathrm{A}$$$$=-987\mathrm{A}+2584\mathrm{I}=\left(\begin{array}{c}1597987\\ 987610\end{array}\right)$$$$$$

Answered by dimentri last updated on 18/Aug/23

$${\left(\begin{array}{c}1-1\\ -12\end{array}\right)}^{-1}=\left(\begin{array}{c}21\\ 11\end{array}\right)=X$$$${\left(\begin{array}{c}1-1\\ -12\end{array}\right)}^{-2}=X^2=\left(\begin{array}{c}21\\ 11\end{array}\right)\left(\begin{array}{c}21\\ 11\end{array}\right)=\left(\begin{array}{c}53\\ 32\end{array}\right)=3X-I$$$${\left(\begin{array}{c}1-1\\ -12\end{array}\right)}^{-3}=X(3X-I)=3X^2-X$$$$=3(3X-I)-X=8X-3I$$$${\left(\begin{array}{c}1-1\\ -12\end{array}\right)}^{-4}=X(8X-3I)=8X^2-3X$$$$=8(3X-I)-3X=21X-8I$$$${\left(\begin{array}{c}1-1\\ -12\end{array}\right)}^{-8}=(21X-8I)(21X-8I)$$$$=441X^2-336X+64I$$$$=441(3X-I)-336X+64I$$$$=987X-377I$$$$=\left(\begin{array}{c}1974987\\ 987987\end{array}\right)-\left(\begin{array}{c}3770\\ 0377\end{array}\right)$$$$=\left(\begin{array}{c}1597987\\ 987610\end{array}\right)$$

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