Matrix Groups in Abstract Algebra

Question

QUESTION: Let $h= \begin{pmatrix} -1 & 1\\-1&0 \end{pmatrix} \in GL_2(\Bbb R)$. Find $\langle h\rangle$.

I'm stuck on the solution, but here is what I have:

Let $h=\begin{pmatrix} -1&1\\-1&0 \end{pmatrix}$. Then $h^2 =\begin{pmatrix}0&-1\\1&-1 \end{pmatrix}$, $h^3 = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}=I$, which is the identity matrix.

I'm not entirely sure where to go from here... Could someone explain how I can get the cyclic group $\langle h \rangle$?

Answer 1

You've found all the elements in $$\langle h\rangle: \{I= h^3, h, h^2\}$$ It's a cyclic subgroup of $GL_2(\mathbb R)$ of order $3$.