Order of $GL(n, \mathbb Z_p)$ [duplicate]

Let $G$ be the group of all $n\times n$ matrices with entries of the matrices from the field $\mathbb Z_p$ , $p$ prime, such that determinant of every matrix is not $[0]$ , w.r.t. matrix multiplication ; then what is the order of $G$ ?

Solution 1:

Further hint:

For any matrix in $\;G\;$ , we look at its columns. The first one cannot be zero, so there are $\;p^n-1\;$ possible choices for it.

Now, the second column must be linearly independent of the first one, so we must avoid all its $\;p\;$ scalar multiples (pay attention to the fact that this already takes care of a possible zeros column), so there are $\;p^n-p=p\left(p^{n-1}-1\right)\;$ possible choices for this second column.

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