Linear algebraic group inside $GL_n$

Question

Let $G$ be a linear algebraic group. Consider the closed embedding

$G \hookrightarrow GL_n$.

Let $K$ be any field. Let $x \in GL_n(K)$. Now suppose we know that $x^n \in G(K) $ for some positive integer $n$. Can we conclude that $ x \in G(K)$? Thanks for help. Is there any counterexample to this?

Answer 1

Consider $G=\mathbb{G}_m\subset GL_2$, embedding as scalar matrices. Let $x$ be the diagonal matrix $[1,-1]$ (and characteristic not 2). Then $x^2\in G(K)$, but $x\not\in G(K)$.