Irreducible Semigroups of Matrices with Eigenvalue One

We study groups and semigroups of n x 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 $ngeq 5$ 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.