Order of Group Element/Examples/Matrix (1 1, 0 1) in General Linear Group

Examples of Order of Group Element
Consider the general linear group $\GL 2$.

Let $\mathbf A := \begin{bmatrix} 0 & 1 \cr 1 & 1 \end{bmatrix} \in \GL 2$

The order of $\mathbf A$ in $\GL 2$ is infinite.

Proof
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then we see that:


 * $\mathbf A^n = \begin{bmatrix} 0 & 1 \cr n & 1 \end{bmatrix}$

and so:
 * $\forall n \in \Z_{>0}: \mathbf A^n \ne \begin{bmatrix} 0 & 1 \cr 0 & 1 \end{bmatrix}$

Hence the result.