Given the block partitioned matrix $$A = \begin{bmatrix} T_y & -T_u \\ -\Lambda\otimes I& I \end{bmatrix},$$ where $T_y$ is nonsingular, $\Lambda$ is square with $\pm 1$ as nonzero entries and $\otimes$ denotes the kronecker product.
Having $T_{u^i}\neq T_{y^i}$ or $\Lambda_{ii} \neq 0$ for all $i$ is enough to ensure that $A$ is nonsingular?
I know that using the Schur complement we can see that $T_y-T_u(\Lambda \otimes I)$ have to be full rank for A to be nonsinguar, but having $T_{u^i}\neq T_{y^i}$ or $\Lambda_{ii} \neq 0$ would be enough to ensure that?