Prove or disprove the following statements: (a) Let A E Mnxn (F). If A is diagonalizable over F, then A has n distinct eigenvalues over F. (b) Let A, BE Maxn (F) and let u and u be both eigenvectors for both A and B with Au = 6u Bu = 5u Av = 107 BU = 30 Then, 3u + 57 is an eigenvector for AB with the corresponding eigenvalue 30. (c) Let A E Mnxn(F). If A is diagonalizable over F and has eigenvalues 1, 12, . . ., An; then A - All is also diagonalizable over F with eigenvalues 0, 12 - 1, 13 - M1 . .., An - 21. (d) Let A E Mnxn(F). If A is diagonalizable over F, then AT is also diagonalizable over F