[tex]x^2-x-1 = 0 \\ \\ A = \begin{pmatrix}-\epsilon+1&\dfrac{1}{\epsilon}\\ \:\:1&\epsilon-1\end{pmatrix}\\ \\ A^2 = \begin{pmatrix}\epsilon^2-2\epsilon+1+\dfrac{1}{\epsilon}&0\\ 0&\epsilon^2-2\epsilon+1+\dfrac{1}{\epsilon}\end{pmatrix}\\ \\ \epsilon^2-\epsilon-1 = 0 \Big|:\epsilon \\ \epsilon-1-\dfrac{1}{\epsilon}= 0\Rightarrow \dfrac{1}{\epsilon}-\epsilon= -1[/tex]
[tex]\Rightarrow \epsilon^2-2\epsilon+1+\dfrac{1}{\epsilon}= \epsilon^2-\epsilon + 1-\epsilon+\dfrac{1}{\epsilon}=0-\epsilon+\dfrac{1}{\epsilon}= -1\\ \\ \\ \Rightarrow A^2 = \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} = -I_2 \\ \\ A^2 = -I_2\Big|^{1009} \\ \\ A^{2018} = (-I_{2})^{1009} \\ \\ A^{2018} = -I_{2} \Big|\cdot A \\ \\ A^{2019} = -I_{2}\cdot A\\ \\ \Rightarrow \boxed{A^{2019} = -A}[/tex]
