Assume A, B, P, and D are n times n matrices. Determine whether the following statements are true or false. Justify each answer.
A matrix A is diagonalizable if A has n eigenvectors.
The statement is false. A matrix is diagonalizable if and only if it has n -1 linearly independent eigenvectors.
The statement is true. A diagonalizable matrix must have more than one linearly independent eigenvector.
The statement is true. A diagonalizable matrix must have a minimum of n linearly independent eigenvectors.
The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.
If A is diagonalizable, then A has n distinct eigenvalues.
The statement is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent eigenvectors.
The statement is true. A diagonalizable matrix must have n distinct eigenvalues.
The statement is false. A diagonalizable matrix must have more than n eigenvalues.
The statement is true. A diagonalizable matrix must have exactly n eigenvalues.
If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.
The statement is true. AP = PD implies that the columns of the product PD are eigenvalues that correspond to the eigenvectors of A.
The statement is false. If P has a zero column, then it is not linearly independent and so A is not diagonalizable.
The statement is true. Let v be a nonzero column in P and let lambda be the corresponding diagonal element in D. Then AP = PD implies that Av = lambda v, which means that v is an eigenvector of A.
The statement is false. AP = PD cannot imply that A is diagonalizable, so the columns of P may not be eigenvectors of A.