Let $\mathbb{F}$ be a finite field.
Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra generated by a subgroup of $\mbox{GL}_n(\mathbf{F})$. 

Equivalently, let $G$ be a subgroup of $\mbox{GL}_n(\mathbf{F})$, and let $A:=\mbox{span}_\mathbb{F}(G)$,
the span as a vector space over the field $\mathbb{F}$.

Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$ 
that one needs to take such that, with high probability, the algebra generated by $a_1,\dots,a_k$ is $A$?

I am interested in a bound that applies to _all_ such $A$. 
Replacing "high probability" by "probability bounded away from zero" would also be fine.

Taking $k=O(n^2)$ will provide a set that spans $A$ as a vector space. 
Would less than that suffice?

**Comment:** This is related to [this question][1].


  [1]: http://mathoverflow.net/questions/154761/finding-a-basis-for-the-linear-combinations-span-of-a-matrix-group-efficientl/206338#206338