This is an extract from a conference paper. It seems the authors are invoking Tutte's theorem (since [12] refers to the 1947 paper) to conclude that a matrix $J(x)$ with given numerical entries is invertible. I have seen it explicitly mentioned that the non-singular determinant condition in Tutte's theorem is not tested numerically but as a polynomial identity. Surely there is a problem here ? If I follow this reasoning here then the matrix $\begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}$ should be invertible ?
Update: The authors state in the paper "However, the invertibility of J(x) depends on x, and is thus hard to characterize. For this reason, we resort to studying the generic rank of J(x), which is the maximal rank over all possible values for the non-zero entries of J(x)."