The Degree of an Exact Order Matrix

The classes of exact order k matrices for any positive integer k, were defined and studied by Mohan, Parthasarathy and Sridhar (Mohan, S. R., T. Parthasarathy, R. Sridhar. 1994. The linear complementarity problem with exact order matrices. Math. Oper. Res.19 618–644.). Here, we prove results on the linear complementarity problem LCP(q, M), for M belonging to the class of exact order k, k ≥ 3, using the concepts of degree theory. Our main result in this paper consists in proving that a matrix M ∈ R^n×n of exact order k, for any positive integer n ≥ k + 3, belongs to the class Q if and only if the degree of M is either +1 or −1. Also, a complete characterization of exact order 2 matrices is presented, in terms of their inverse structure.