Eventually r-cyclic matrices are defined, and it is shown that if A is an eventually r-cyclic matrix A having rank A2 = rank A, then A is r-cyclic with the same cyclic structure. This result and known Perron-Frobenius theory of eventually nonnegative matrices are used to establish an algorithm to determine whether a matrix is strongly eventually nonnegative (i.e., is an eventually nonnegative matrix having a power that is both irreducible and nonnegative).