We consider the linear complementarity problem (LCP), w = Az+q, W grea
ter-than-or-equal-to 0, Z greater-than-or-equal-to 0, W(T)Z = 0, when
all the off-diagonal entries of A are nonpositive (the class of Z-matr
ices), all the proper principal minors of A are positive and the deter
minant of A is negative (the class of almost P-matrices). We shall cal
l this the class of F-matrices. We show that if A is a Z-matrix, then
A is an F-matrix if and only if LCP(q, A) has exactly two solutions fo
r any q greater-than-or-equal-to 0, q not-equal 0, and has at most two
solutions for any other q.