I want to prove that the equation \begin{align}u^3-2u^2+uv+\operatorname{Id}=0\end{align} has a solution for $|v|$ small enough. My goal is to use Implicit Function Theorem. I assume that the matrices are of $n\times n$ dimension.
Implicit Function Theorem states that
Let $U$ be open in $\Bbb{R}^n,$ $V$ open in $\Bbb{R}^m$. Let \begin{align}f:U\times V\to \Bbb{R}^m\end{align} \begin{align}(x,y)\mapsto f(x,y)\end{align} be a $C^K$ function. Assume there exists $(a,b)\in U\times V$ for which $f(a,b)=0$ and $D_y f(a,b)\in GL(\Bbb{R}^m).$ Then, there exists $U_1\subset U$ an open neighborhood of $a$ in $\Bbb{R}^n$, $U_2\subset V$ an open neighborhood of $b$ in $\Bbb{R}^m$ and a $C^K$ function $\varphi:U_1\to U_2$ such that \begin{align}f(x,y)=0\leftrightarrow y=\varphi(x)\end{align} for $(x,y)\in U_1\times U_2.$
Please, how do I go about this? Any help?