How many invertible \(n\times n\) matrices are there over \(\FF_p\)?
Solution
An invertible matrix is equivalent to an ordered basis of the vector space \(\FF_p^n\).
Choose the columns successively. The first column can be any nonzero vector: \[ p^n-1 \] choices. After choosing \(k\) linearly independent columns, their span has \(p^k\) elements, so the next column has \[ p^n-p^k \] choices.
Therefore \[ |GL_n(\FF_p)| = (p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}). \] Equivalently, \[ |GL_n(\FF_p)| =p^{n(n-1)/2}\prod_{j=1}^n(p^j-1). \]