Let $p$ be prime, let $A$ be the $n\times n$ companion matrix associated to a primitive polynomial $f(x)$ of degree $n$ with coefficients in ${\mathbb Z}_p$, and let $T=\{A^j\mid 1\leq j\leq p^n-1\}\subseteq {\mathbb Z}_p^{n\times n}$. Note that $T$ can be identified with the nonzero elements of $GF(p^n)$. Suppose $S$ is the subset of $T$ consisting of matrices possessing the property that every leading principal minor is nonzero. A couple of questions: (1) What is the cardinality of $S$? and (2) Does $S$ have some special meaning as a subset of $GF(p^n)$ or otherwise? Perhaps (1) depends on $f(x)$.
Thanks for your thoughts!