In this paper, we study properties of the solution of the pseudomonotone second-order cone linear complementarity problems (SOCLCP). Based upon Tao’s recent work [Tao, J. Optim. Theory Appl., 159(2013), pp. 41–56] on pseudomonotone LCP on Euclidean Jordan algebras, we made two noticeable contributions on the solutions of the pseudomonotone SOCLCP: First, we introduce the concept of Jn-eigenvalue of a matrix, and prove that the associated matrix of the pseudomonotone SOCLCP always admits a Jn-eigenvalue; the notion of the Jn-eigenvalue turns out be a key for the pseudomonotone SOCLCP as it not only generalizes the results on the SOCLCP with the globally uniquely solvable (GUS) property, but also offers insight into the corresponding LCP and lays a foundation for computing the solution of SOCLCP. Second, we perform a thorough analysis of the range of the pseudomonotone SOCLCP, and establish an explicit and complete description of the range of the SOCLCP.