  We establish a characterization of almost $P$-matrices via a sign
non-reversal property. In this we are inspired by the analogous results for
$N$-matrices. Next, the interval hull of two $m \times n$ matrices $A=(a_{ij})$
and $B = (b_{ij})$, denoted by $\mathbb{I}(A,B)$, is the collection of all
matrices $C \in \mathbb{R}^{m \times n}$ such that each $c_{ij}$ is a convex
combination of $a_{ij}$ and $b_{ij}$. Using the sign non-reversal property, we
identify a finite subset of $\mathbb{I}(A,B)$ that determines if all matrices
in $\mathbb{I}(A,B)$ are $N$-matrices/almost $P$-matrices. This provides a test
for an entire class of matrices simultaneously to be $N$-matrices/almost
$P$-matrices. We also establish analogous results for semipositive and
minimally semipositive matrices. These characterizations may be considered
similar in spirit to that of $P$-matrices by Bialas-Garloff [Linear Algebra
Appl. 1984] and Rohn-Rex [SIMAX 1996], and of positive definite matrices by
Rohn [SIMAX 1994].
