Hw5 Problem 3

Describe all of the elements in the cyclic subgroup of $GL(2,\ \mathbb{R})$ generated by the matrix $\begin{bmatrix} 0 & -2 \\ -2 & 0 \end{bmatrix}$.

Solution:

$\bigg \langle \begin{bmatrix} 0 & -2 \\ -2 & 0 \end{bmatrix} \bigg \rangle$ consists of all elements of the form $\begin{bmatrix} 4^{n} & 0 \\ 0 & 4^{n} \end{bmatrix}$ or the form $\begin{bmatrix} 0 & (-2)^{2n+1} \\ (-2)^{2n+1} & 0 \end{bmatrix}$ where $n \in \mathbb{Z}$.