Which of the following statements are always true need helpm

Which of the following statements are always true ?...need help...many thanks !!

Let C be an n times n matrix. Which of the following statements are always true? Statement A: The matrix C can be diagonalised only if C has n distinct eigenvalues. Statement B: The matrix C can be diagonalised only if C has n linearly independent eigenvectors. Only A Both A and B Neither A nor B Only B

Solution

A square n x n matrix A is diagonalizable if there is a diagonal matrix D and an invertible matrix P such that P^-1 AP = D. Here, the matrix D has eigenvalues of A on its main diagonal and all other elements of D are 0. Also, P has columns which are the corresponding eigenvectors of A (arranged in the same order as that of the eigenvalues on the main diagonal of D). Since P has to be invertible, its columns have to linearly independent ( otherwise det (P) = 0 and P is not invertible). There is no necessity of all distinct eigenvalues as the same eigenvalues can have distinct eigenvectors. Also, if a square matrix A has certain eigenvalues of certain multiplicity, then also P can be invertible. Thus only statement B is true.