For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth.

Question

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth. So where I am starting is by extending $Ad : G \rightarrow GL_d(\mathbb{R})$ to $Ad : GL_d(\mathbb{R}) \rightarrow GL_d(\mathbb{R})$, which if I can show is smooth will let me know that the former is smooth. And from my text I know that $Ad_g(B) = gBg^{-1}$ So if i replace B with $E_{ij}$ as a standard basis. I can show that from cramers rule, that all entries will be rational. Am I on the right track?