Alternating groups and linear groups

Question

I learned from the literature that the symmetric group $S_n$ can be viewed a subgroup of permutation matrices of $GL_n(q)$ (where $q$ is a prime power), the general linear group of invertible $n\times n$ matrices over a field of order $q$. Therefore $A_n$ can be viewed as a subgroup of $SL_n(q)$. Can you help me to understand what happens when we go to $PSL_n(q)$? Is $A_n$ still a subgroup of $PSL_n(q)$?