Some sign patterns that allow a real inverse pair B and B−1 Original Research Article

A sign pattern matrix is a matrix whose entries are from the set {+, −, 0}. For a real matrix B, by sgn B we mean the sign pattern matrix in which each positive (negative, zero) entry is replaced by + (−, 0). If A is an n-by-n sign pattern matrix, then the sign pattern class of A is defined by Q(A) = {B set membership, variant Mn(R)sgn B = A}. Our purpose here is to investigate patterns that allow some B and B−1 to be in Q(A). To this end, we establish global necessary conditions, we obtain necessary and sufficient conditions for certain classes of patterns, and we provide several construction algorithms to obtain classes of patterns that have the inverse pair property.