Subgroups of General Linear Group I have a question about the order of the subgroups of $GL_2(\mathbb{R})$ generated by the following matrix: 
$\begin{pmatrix}
 1&-1 \\ 
 -1&0 
\end{pmatrix}$
So I computed several powers of this matrix and realized that the order of the subgroup generated by this matrix is infinity. However, I do not know how to prove it in general that:
${\begin{pmatrix}
 1&-1 \\ 
 -1&0 
\end{pmatrix}}^n\neq \begin{pmatrix}
 1&0 \\ 
 0&1 
\end{pmatrix} \forall n\in\mathbb{Z}.$
Is there any way to prove this in general? I might have did not remember something from Linear Algebra to prove this. Any suggestion is really appreciated.
Here are what I got for my calculations: 
${\begin{pmatrix}
 1&-1 \\ 
 -1&0 
\end{pmatrix}}^2= \begin{pmatrix}
 2&-1 \\ 
 -1&1 
\end{pmatrix} $
${\begin{pmatrix}
 1&-1 \\ 
 -1&0 
\end{pmatrix}}^3= \begin{pmatrix}
 3&-2 \\ 
 -2&1 
\end{pmatrix} $
${\begin{pmatrix}
 1&-1 \\ 
 -1&0 
\end{pmatrix}}^4= \begin{pmatrix}
 5&-3 \\ 
 -3&2 
\end{pmatrix} $
${\begin{pmatrix}
 1&-1 \\ 
 -1&0 
\end{pmatrix}}^5= \begin{pmatrix}
 8&-5 \\ 
 -5&3 
\end{pmatrix} $
${\begin{pmatrix}
 1&-1 \\ 
 -1&0 
\end{pmatrix}}^6= \begin{pmatrix}
 13&-8 \\ 
 -8&5 
\end{pmatrix} $

Hint: When you diagonalize this matrix, you get 
$
S\begin{pmatrix} \frac{-1 + \sqrt{5}}{2} & 0 \\ 0 & \frac{1-\sqrt{5}}{2} \end{pmatrix}S^{-1}$
for some ugly matrices $S$ and $S^{-1}$. These numbers are larger in norm than $1$.
