  A real square matrix $A$ of order $n \times n~ (n \geq 3)$ is called an
$F_0$-matrix, if it is a $Z$-matrix (off-diagonal entries nonpositive), all of
whose principal submatrices of orders at most $n-2$ are $M$-matrices while
there is at least one principal submatrix of order $n-1$, which is an
$N_0$-matrix. An $M$-matrix is a $Z$-matrix with the property that the real
parts of all its eigenvalues are nonnegative. An $N_0$-matrix, in turn, is
characterized by the fact that it is an invertible $Z$-matrix whose inverse is
(entrywise) nonpositive. The first aim of this article is to present a short
survey of some subclasses of $Z$-matrices, pertinent to the second objective,
where new results concerning $F_0$-matrices are presented.
