Please explain and justify each answer 3. Let A, P, D are n x n matrices. Mark each statement True or False. Justify each answer.a) A is diagonalizable if A = PDP- for some matrix D and some invertible matrix P.b) If R" has a basis of eigenvectors of A, then A is diagonalizable.c) A is diagonalizable if A has n eigenvectors.

Given: Let A,D,P are n×n matrices a) A is diagonalizable if A=PDP-1 for some matrix D and some invertible matrix P This statement is False. A n×n matrix is diagonalizable if it can be written in the form A=PDP-1 where D is a diagonal n×n matrix with the eigenvalues of A and P is a invertible matrix b)If ℝn has a basis of eigenvectors of A,then A is diagonalizable. This statement is True. An n-by-n matrix A over the field F is diagonalizable if and only if the sum of the dimensions of itseigen spaces is equal to n, which is the case if and only if there exists a basis of F n consisting of eigenvectors of A. If such a basis has been found, one can form the matrix P having these basis vectorsas columns, and P-1AP will be a diagonal matrix. The diagonal entries of this matrix are theeigenvalues of A.c) A is diagonalizable if a A has n eigenvectors. This is statement is False. By the Diagonalization theorem, An m×m matrix A is said to be diagonalizable if and only if A has m linearly independent eigenvectors of A.