In my course on multivariate calculus we treat the implicit function theorem and I am stuck on the following question:
Find the values of $a$ and $b$ such that, in a neighbourhood of $(x,y,u,v) = (0,1,1,-1)$, $x$ and $y$ are implicitly defined as $C^1$-functions $f$ and $g$ of $u$ and $v$ by the system of equations : $$\begin{cases} e^x +uy + u^4v-a=0 \\ y\cos(x) +bx +b^2u -2v = 4 \end{cases}$$
NB: check whether all conditions of the implicit function theorem are satisfied and state the conclusions as accurately as possible.
My work so far:
Let $G : \mathbb{R}^4 \to \mathbb{R}^2$ be defined by: $$G(x,y,u,v)= \begin{pmatrix} e^x +uy + u^4v -a\\ y\cos(x) +bx +b^2u -2v \end{pmatrix} $$ Obviously, this function is $C^1$.
After this, I presume I have to form a matrix of partial derivatives and check for what values of $a$ and $b$ that matrix is non-singular.
Could anyone give me some guidance on how to continue?
Thanks in advance.