So I have this two equations, $$xz^3+yu+ax=1$$ $$2xy^3+u^2z+a(y-1)=0$$ that define (x,y) as an implicit function of (z,u) in a neighbourhood of the point $(x_0,y_0,z_0,u_0)=(0,1,0,1)$. Now, if we call $(x,y)=G(u,z)$, I need to know for which values of a, G admits a local inverse of $C^1$ class in a neighbourhood of the point $(0,1)$. My procedure is:
Using the implicit function theorem, we know that $$\text{Jacobian}G(0,1)=-\begin{pmatrix} a & 1 \\ 2 & a \\ \end{pmatrix}^{-1} \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}=\begin{pmatrix} \frac{1}{a^2-2} & \frac{-a}{a^2-2} \\ \frac{-a}{a^2-2} & \frac{2}{a^2-2} \\ \end{pmatrix} $$ Which determinant is $\frac{2-a^2}{a^2-2}=0 \Leftrightarrow a=\pm \sqrt{2}$, so in conclusion, G admits local inverse for every value of a except $\pm \sqrt{2}$
I want to know if my solution is correct.