Suppose there is a matrix that satisfies the polynomial . Prove that 3 divides n.
If then the minimum polynomial of A must be a factor of . But that polynomial is irreducible over the rationals. Therefore p(x) must be the minimum polynomial of A. The factors of the characteristic polynomial of A must be the same as those of the minimum polynomial. But again, since p(x) is irreducible over , it only has one factor, namely itself. Therefore the characteristic polynomial must be of the form for some integer k. This has degree 3k, and so , as required.
That naturally raises the question of whether there are any such matrices. In fact there are, and you can check that satisfies p(B)=0. The most general 3x3 matrix satisfying the equation would then be similar to B. In other words, it would be of the form , where P is any invertible matrix in . In general, any matrix satisfying the equation would be similar to a direct sum of copies of B.