  In this paper, we consider matrices whose inverses are tridiagonal
Z--matrices. Based on a characterization of symmetric tridiagonal matrices by
Gantmacher and Krein, we show that a matrix is the inverse of a tridiagonal
Z--matrix if and only if, up to a positive scaling of the rows, it is the
Hadamard product of a so called weak type $\D$ matrix and a flipped weak type
$\D$ matrix whose parameters satisfy certain quadratic conditions. We predict
from these parameters to which class of Z--matrices the inverse belongs to. In
particular, we give a characterization of inverse tridiagonal M--matrices.
Moreover, we characterize inverses of tridiagonal M--matrices that satisfy
certain row sum criteria. This leads to the cyclopses that are matrices
constructed from type $\D$ and flipped type $\D$ matrices. We establish some
properties of the cyclopses and provide explicit formulae for the entries of
the inverse of a nonsingular cyclops. We also show that the cyclopses are the
only generalized ultrametric matrices whose inverses are tridiagonal.
