Why we need to study representations of matrix groups? For example, the group $SL_2(F_q)$, where $F_q$ is the field with $q$ elements, is studied by Drinfeld. I think that these groups are already given by matrices. The representation theory is to represent elements in an algebra or group (or other algebraic structure) by matrices. Why we still need to study representations of matrix groups? Thank you very much.

Why we need to study representations of matrix groups? For example, the group $\operatorname{SL}_2(\mathbb F_q)$, where $\mathbb F_q$ is the field with $q$ elements, is studied by Drinfeld. I think that these groups are already given by matrices. The representation theory is to represent elements in an algebra or group (or other algebraic structure) by matrices. Why we still need to study representations of matrix groups? Thank you very much.