15. (5%) For square matrices A and B, we say that B is similar to A if there exists an invertible matrix P such that B = AP.
Which of the following statements about similarity is NOT always correct?
(A) If A and B are similar and A is singular, then B is singular
(B) If A and B are similar and A is invertible, then B is invertible
(C) If A and B are similar. then A and B have the same range
(D) If A is invertible, then AB is similar to BA
(E) If A and B are similar, then A and B may or may not have the same eigenvalues