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.