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What is the number of invertible $n \times n$ matrices with entries in a field $k$ with $q$ elements?

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    https://en.wikipedia.org/wiki/General_linear_group "over finite fields" – pjs36 Sep 15 '15 at 17:56
  • See my answer here: http://math.stackexchange.com/questions/1353991/determining-the-cardinality-of-sl-2-f-3/1354022#1354022 – Matt B Sep 15 '15 at 18:17
  • How on earth does this have 4 upvotes? No context, and a question that has been asked multiple times previously. – Viktor Vaughn Sep 15 '15 at 21:03

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The row vectors of such a matrix have to be linear independent. For the first row we have $q^n-1$ possibilities. For the next row there are $q^n-q$ left and so on. So we get for the total number $$\prod_{i=0}^{n-1} (q^n -q^i) = q^{\frac{n(n-1)}{2}} \prod_{i=1}^{n} (q^i -1).$$

principal-ideal-domain
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