(a) Suppose that A is a 3 x 3 diagonalizable matrix and that { v1, v2, v3} is a basis of R' consisting of eigenvectors of A with corresponding eigenvalues 1, 12, 13. Suppose that the matrix B is also a 3 x 3 diagonalizable matrix with the same eigenvectors as A, although with possibly different eigenvalues 1, 62, 63. Prove that A + B + 21 is a diagonalizable matrix, where I is the 3 x 3 identity matrix. (b) Let C be a 4 x 4 matrix of rank 3. If -1, 2, and 7 are eigenvalues of C, show that C' is diagonalizable. (c) Suppose that X is an n x n diagonalizable matrix. Show that X is diagonalizable