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Possible Duplicate:
Groups/Linear maps

Given a natural number $n$, consider the set of all $n\times n$ matrices where each element is a member of $\mathbb Z_p$, where $p $ is a prime.

How many of these matrices are invertible modulo $p$?

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marked as duplicate by Arturo Magidin, Dylan Moreland, azarel, Willie Wong Mar 9 '12 at 11:52

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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What you're looking for is Theorem 2.1.1 in the following PDF notes: $$ | {\rm GL}(n, \mathbb{Z}_p) | = \prod_{i=0}^{n-1} (p^n- p^i) $$

You can learn more about ${\rm GL}(n,\mathbb{Z}_p)$ here on Wikipedia.

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