Given a continuously differentiable function $f:\Bbb R^2\to \Bbb R^2$ so that for all $x\in \Bbb R^2$ there is $N_x\in \Bbb N$ and $2\times2$ invertible matrix $A_x$ so that $[Df_x]^{N_x}=A_x\begin{bmatrix}2&1\\5&5\\ \end{bmatrix}A_x^{-1}$. Prove that for any $x\in \Bbb R^2$, there exists an open neighborhood $U$ so that $f:U\to f(U)$ is invertible.
I think I should use inverse function theorem to solve it. So I try to calculate $$\det([Df_x]^{N_x})=\det(A_x\begin{bmatrix}2&1\\5&5\\ \end{bmatrix}A_x^{-1})=\det(\begin{bmatrix}2&1\\5&5\\ \end{bmatrix})=5\neq0$$, and $\det([Df_x]^{N_x})=\det([Df_x])^{N_x}$. So the Jacorbian determinant is not $0$. By the inverse function theorem, there exist neighborhood $U_x$ for $x$ and $V_x$ for $f(x)$ such that $f(U_x)=V_x$ and $f$ is invertible on $U_x$.
Do I do all what this want me to do? Thanks for any suggestions.