The minimum upper bound on the first ambiguous power of an irreducible, nonpowerful ray or sign pattern

Let A be an n×n irreducible ray or sign pattern matrix. If A is a sign pattern, it is shown that either A is powerful or else Ak has an ambiguous entry for some , and further, sign patterns based on the Wielandt graph show that this bound is the best possible. If A is a ray pattern, partial results for the same bound are given.