Subgroups of General Linear Group

Question

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} $

Answer 1

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 $1$.

Answer 2

Hint: if $\lambda$ is an eigenvalue of a matrix $A$ (that is, there exists a nonzero column vector $v$ such that $Av = \lambda v$), then $\lambda^n$ is an eigenvalue of the matrix $A^n$.

Suppose that $A^n = I$ for some $n$. What are the eigenvalues of $I$?