Let A, P, and D by nxn matrices. Mark each statement true or false. Justify each answer. Complete parts (a) through (d) below. a. A is diagonalizable if A PDP 1 for some matrix D and some invertible matrix P. Choose the correct answer below. O A. The statement is true because the columns of P are the n linearly independent eigenvectors of A. O B. The statement is false because A must have n distinct eigenvalues for the matrix D and P to exist. O C. The statement is true because ifA -PDP-1, then A is diagonalizable by definition. O D. The statement is false because the symbol D does not automatically denote a diagonal matrix. b. If R" has a basis of eigenvectors of A, then A is diagonalizable. Choose the correct answer below. O A. The statement is true because A is diagonalizable if and only if there are enough eigenvectors to form a basis of IRn O B. The statement is false because the eigenvectors of A must also be linearly independent. O C. The statement is true because all square matrices are diagonalizable. OD. The statement is false because A must also have n distinct eigenvalues A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. Choose the correct answer below. ? A. The statement is false because it also required that A is invertible ?B The statement s rue because A = PDP 1 where the diagonal entries of D are eigenvalues of A hat correspond respect e O C. The statement is false because the eigenvalues of A may not produce enough eigenvectors to form a basis of R" O D. The statement is true because there are n corresponding eigenvectors that form a basis of IR to the e en ec s np d.