Let $\mathbb{F}_q$ be the finite field with $q$ elements. I want to calculate the number of elements of the set $S=\{A=(a_{ij}) \in Gl_n(\mathbb{F}_q):a_{11}=1\}.$
We know that $|Gl_n(\mathbb{F}_q)|=(q^n-1)(q^n-q)\cdots(q^n-q^{n-1}).$ Trying to define some group homomorphism from $Gl_n(\mathbb{F}_q)$ to $\mathbb{F}_q^{\times}$ whose kernel is $S,$ but failed to do so. Any help will be appreciated. Thanks.