Question
Let $A$ be a central simple algebra over $F$ where $F$ is a field and $dim_FA=n^2$.
$A^\times$ can be seen as an algebraic group over $F$ and let $G=A^\times$.
Prove that $G$ is an inner form of $GL_n$.
I list what I know below.
$(1)$ Since $\overline{F}$ (the algebraic closure of $F$) is a splitting field for $A$, there is an isomorphism $\iota :A_\overline{F}=A\otimes_F\overline{F}\rightarrow M_n(\overline{F})$. This induces an isomorphism $\xi :G(\overline{F})\rightarrow GL_n(\overline{F})$. Hence $G$ is a $F$-form of $GL_n$.
$(2)$ To solve the question, we have to show that for all $\sigma\in Gal(\overline{F}/F)$, $c_\sigma=\xi^{-1} \circ\sigma(\xi) \in IntG$ where $\sigma(\xi)=\sigma(\xi(\sigma^{-1}(x))) (x\in G(\overline{F}))$.
I've tried to prove it with the Skolem-Noether theorem but I failed.
How to prove it?