Let $M_n(F)$ denote the set of $n\times n$ matrices for a value $n\in \mathbb{N}$ with components in a field $F$ with finite cardinality . Let $$I=\{A\in M_n(F):~~ \det(A) \neq 0 \}.$$ What is the cardinality of $I$? I'm looking for a function $f: \mathbb{N} \rightarrow \mathbb{N}$ that takes in the rank $n$ and maps it to the cardinality of $I$ which is discrete.
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$\begingroup$ see Arithmetic of Finite Fields by Charles Small. $\endgroup$– Will JagyCommented 4 hours ago
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$\begingroup$ Eye eye Captain Willy 🫡. $\endgroup$– Wuu tang clanCommented 4 hours ago
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2$\begingroup$ $$\prod_{k=0}^{n-1}(q^n-q^k)\text{ where }q=|F|$$ $\endgroup$– bofCommented 4 hours ago
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1$\begingroup$ I don't have access because I'm not at institution anymore but will try to find free version somewhere. It was published 1991 should be free pdf somewhere 😂 $\endgroup$– Wuu tang clanCommented 3 hours ago
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1$\begingroup$ I think this answer also resolves your question: math.stackexchange.com/a/73963 $\endgroup$– Martin SkilleterCommented 2 hours ago
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