Suppose there is a matrix that satisfies the polynomial . Prove that 3 divides n.
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.