please help me on this question thank you ps it is ture or f

please help me on this question, thank you ps: it is ture or false not multiple choice

A, P and D are n times n matrices. Check the true statements below: A. A is diagonalizable it A = PDP^-1 Tor some diagonal matrix D and some invertible matrix P. B. A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. C. If there exists a basis for R^n consisting entirely or eigenvectors or A, then A is diagonalizable. D. If A is diagonalizable, then A is invertible.

Solution

A. The statement being the definition nof a diagonalizable matrix A, is True.

B. The statement is False. A is diagonalizable if and only if A has n linearly independent eigenvectors.

C. The statement implies that A has n linearly independent eigenvectors.. Hence A is diagonalizable. The statement is true.

D. The stamen is False. There is no relation between invertibility and diagonalizability.