On the matrix class Q0 and inverse monotonicity properties of bordered matrices

Let A be a square matrix satisfying either a monotonicity condition (like inverse nonnegativity) or a property expressed in terms of the existence of solution of the linear complementarity problem, defined using A (like the Q0-property). We address the question of when the bordered matrix M, whose leading (maximal) proper principal submatrix being the matrix A, inherits the said property of A.