Suppose $\psi: GL_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ defined by sending $$ g\mapsto gAg^{-1}. $$
Then why is it that $d\psi:T_eGL_2(\mathbb{C})=M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ is defined to be $$ C\mapsto [C,A]? $$
It is sort of related to this link, but I am not sure if the same strategy as in that link will work here.