  A square matrix $M$ with real entries is said to be algebraically positive
(AP) if there exists a real polynomial $p$ such that all entries of the matrix
$p(M)>0$. A square sign pattern matrix $S$ is said to allow algebraic
positivity if there is an algebraically positive matrix $M$ whose sign pattern
class is $S$. On the other hand, $S$ is said to require algebraic positivity if
any matrix $M$, having sign pattern class $S$, is algebraically positive.
Motivated by open problems raised in the work of Kirkland, Qiao and Zhan (2016)
on AP matrices, we list down all nonequivalent irreducible $3\times 3$ sign
pattern matrices and classify each of them into three groups (i) those that
require AP, (ii) those that allow but not require AP, or (iii) those that do
not allow AP. We also give a necessary condition for an irreducible $n\times n$
sign pattern to allow algebraic positivity.
