For a set of non liner equations $f_i(\vec x)$ were $i\in \mathbb{N}$ was the index, one can construct a vector \begin{equation} F(\vec x)= \begin{pmatrix} f_1{\vec x} \\ f_2(\vec x) \end{pmatrix} \end{equation} from there one could use the Gaussian method to calculate the roots \begin{equation} \vec{x}_{n+1} - \vec{x}_n=(J(F(\vec x)) )^{-1} F(\vec x) \end{equation} However, in a question, it was almost impractical to compute the inverse of the Jacobian, that the numerical inverse of the jacobian matrix were not identity matrix \begin{equation} \max( (J(F(\vec x)) )^{-1} J(F(\vec x)) )>>1 \end{equation} , but with large numerical errors.
Thus, is it possible to somehow use the Gaussian method without using the matrix inverse? i.e. calculate it more in the sense of the traditional Gaussian method with scalar inputs.
How to compute the Gaussian method without using the matrix inverse?