A banded matrix is a real square matrix where nonzero entries appear around the main diagonal. In this article, we consider linear complementarity properties of (variants) of banded matrices. Focusing on triangular matrices and the newly defined bidiagonal southwest matrices, we describe several results characterizing the Q-property in terms of the sign patterns and determinant of the given matrix. As a byproduct, we describe all Q-matrices of size 2 by 2. Extending these results to Euclidean Jordan algebras, we consider matrix-based linear transformations and study the Q-property. In particular, we show that a rank-one linear transformation of the form a b has the Q-property if and only if either a>0, b>0, or a<0, b<0.
Pratihar et al. (Wed,) studied this question.