This problem establishes a special case of the Cayley-Hamilton theorem. a) Prove that if B is a (3 x 3) matrix and if Bx = 0 for every x in R^3, then B is the zero matrix. [Hint: Consider Be1, Be2, and Be3.] b) Suppose that λ1, λ2, and λ3 are the eigenvalues of a (3 x 3) matrix A, and suppose that u1, u2, and u3 are corresponding eigenvectors. Prove that if {u1, u2, u3} is a linearly independent set, and if p(t) is the characteristic polynomial of A, then p(A) is the zero matrix.