Order of an element in general linear group Let $A=GL(n,2)$ be the general linear group of size $n$ over the finite field 
$\mathbb{F}_2=\{0,1\}$. For small $n=3,4,5$ etc., I observed any matrix in $A$ has of the order less than or equal to $2^n-1$. Is there any proof of this result?  

Consider $g\in GL(n,2)$ and the sequence $\{g^k\}_{k=0}^\infty$. The order of $g$ is the first index $k$ such that there is some $h<k$ such that $g^h=g^k$. Therefore the set $\{g^n\,:\, n\in\Bbb N\}$ contains exactly $\operatorname{ord}g$ distinct matrices. These matrices are all non-zero elements of the $\Bbb F_2$-algebra $\Bbb F_2[g]$. By Cayley-Hamilton theorem, $\dim_{\Bbb F_2}\Bbb F_2[g]\le n$, and therefore $$\operatorname{ord}g\le\lvert\Bbb F_2[g]\setminus\{0\}\rvert\le 2^n-1$$
In general, we may summarise this discussion in the inequality

$$\operatorname{ord}g\le q^{\dim\Bbb F_q[g]}-1$$

