Let $\mathbb{F}$ be a finite field.
Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra.

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 $A$?
What if $A$ is generated by a _group_ of matrices?

I am interested in a bound that applies to _all_ such $A$. 

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