2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/231224A new class of sign-symmetric matrices is introduced in this paper. Such matrices are named J--sign-symmetric. The spectrum of a J--sign-symmetric irreducible matrix is studied under assumptions that its second compound matrix is also J--sign-symmetric and irreducible. The conditions, when such matrices have complex eigenvalues on the largest spectral circle, are given. The existence of two positive simple eigenvalues $λ_1 > λ_2 > 0$ of a J--sign-symmetric irreducible matrix A is proved under some additional conditions. The question, when the approximation of a J--sign-symmetric matrix with a J--sign-symmetric second compound matrix by strictly J--sign-symmetric matrices with strictly J--sign-symmetric compound matrices is possible, is also studied in this paper.24 pagesSpectral Theory15A48; 15A18, 15A75text