Question

exercitiul 18 va rog​

Answer 1

$A=\begin{pmatrix}1&-1\\ \:1&1\end{pmatrix}\\ \\ A^2 = \begin{pmatrix}0&-2\\ 2&0\end{pmatrix}\Big|^4 \\ \\ A^8 = \begin{pmatrix}0&-2\\ 2&0\end{pmatrix}^2\cdot \begin{pmatrix}0&-2\\ 2&0\end{pmatrix}^2 = \begin{pmatrix}-4&0\\ 0&-4\end{pmatrix}\cdot \begin{pmatrix}-4&0\\ 0&-4\end{pmatrix} = \begin{pmatrix}16&0\\ 0&16\end{pmatrix}\\ \\ A^{10} = \begin{pmatrix}16&0\\ 0&16\end{pmatrix}\cdot \begin{pmatrix}0&-2\\ 2&0\end{pmatrix} = \begin{pmatrix}0&-32\\ 32&0\end{pmatrix}$

$\\ \\ \Rightarrow \dfrac{x_{10}^2+y_{10}^2}{x_8^2+y_8^2} = \dfrac{0+(-32)^2}{16^2+0} = \Big(\dfrac{32}{16}\Big)^2 = 2^2 = 4$