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How do I prove this? a) How many n x n matrices are there with entries chosen from the numbers 0,1,...,q-1?

b) Let q be a prime. How many matrices as in (a) have a determinant that is not divisible by q? (In other words how many nonsingular matrices over the q-element field are there?)

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    What are your thoughts on the problem? What have you tried so far? – 2017-01-16
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    @ Omnomnomnom This is a little beyond my level, just like when I started learning the piano while practicing the basic, I also wanted to emulate some very difficult piece by painstakingly copying every stroke of the key and tone. Frankly, I have no clue where to start, although, I do know that this is a counting problem. say an n x n matrice has n rows, and each of the entry of the rows $n^{q-1}$ possible choices, then each of the column also has $n^{q-1}$ choices? Please give me some hint to start with, or some examples. – 2017-01-16
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    Sure. Just keep in mind that on this site, we expect askers to [provide some context](http://meta.math.stackexchange.com/questions/9959/how-to-ask-a-good-question) with their question. For example, it would have helped if a little bit of that comment was there with your question. – 2017-01-16
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    @ Omnomnomnom I will type up more of my context later tonight when I have more time to think of this question over! thank! – 2017-01-16
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    you don't need all that much; what you put into your comment is enough. The fear is that people might otherwise "dump their homework questions" onto the site without thinking about them. Clearly, this is not what you're doing. – 2017-01-16
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    The first part of your Question is an easy counting problem. For each of the $n^2$ entries you can choose from $q$ values. The second part is more difficult but has been discussed here previously. I will find you a previous Question on that. – 2017-01-16

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Hint (Part 1): If we were to construct such a matrix entry-by-entry, we would select one out $q$ possibilities for every entry. That is, we can uniquely construct each matrix by making $n^2$ independent choices of one out of $q$ different possibilities.

For example: convince yourself that with $q = 3$ and $n = 2$, there are $3^{2^2}$ possible matrices.

Hint (Part 2): This part is a lot trickier. Note that for $i = 1,\dots,n$, the $i$th column must be outside the span of the first $i-1$ columns. That is, we must choose an $i$th column that lies outside of an $(i-1)$-dimensional subspace of $\Bbb F_q^n$.