A matrix $$\begin{bmatrix}w &x \\\ y &z\end{bmatrix}\in\mathbb Z^{2\times 2}$$ is unimodular if $$|wz-xy|=1$$ holds.



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> Is there a parametrization of such matrices with $2wy>(wz+xy)$ and $2xz>(wz+xy)$ with $wz-xy=1$?
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On a criterion for unimodular matrix
The OP clarified in a comment that the variables $w,x,y,z$ were meant to be nonnegative. Under this restriction, the pair of inequalities
$$2wy>wz+xy\qquad\text{and}\qquad 2xz>wz+xy$$
has no solution. Indeed, these inequalities feature nonnegative numbers on both sides, hence multiplying them yields
$$4wxyz>(wz+xy)^2.$$
Rearranging, we get
$$0>(wz-xy)^2,$$
which is a contradiction.


