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?

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