Abstract
We study groups and semigroups of n × n matrices with the property that each matrix has a fixed point, i.e., 1 is an eigenvalue of each matrix. We show that for n = 3 and n ≥ 5 there are irreducible matrix groups and irreducible semigroups of nonnegative matrices with this property. In fact, for n = 3 we determine the structure of any such semigroup. We also present additional hypotheses implying reducibility.
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