The order of a general linear group over a finite field is bounded above

Let $F$ be a field. If $|F| = q$ is finite show that $|GL_n(F)| < q^{n^2}$.

Clearly $GL_n(F)$ is contained in the set of all $n \times n$ matrices over $F$. There are at most $q^{n^2}$ such matrices, so $|GL_n(F)| \leq q^{n^2}$. Moreover, the zero matrix is not in $GL_n(F)$, so the inequality is strict.