Sign idempotent sign pattern matrices that allow idempotence

It is well known that not all sign idempotent sign pattern matrices allow idempotence. In this paper, we found a necessary and sufficient condition for a sign idempotent sign pattern matrix to allow idempotence. In addition, we investigated the nonnegativity of sign idempotent sign pattern matrices that allow idempotence. Finally, we show that if A is a sign idempotent sign pattern matrix, then A   allows idempotence if and only if |A|=[|aij|]|A|=[|aij|] also allows idempotence.