Let A, P, and D by n x n matrices. Mark each statement true or false. Justify each answer.

a. A is diagonalizable if A = PDP^-1 for some matrix D and some invertible matrix P. Choose the correct answer below.

O The statement is true because the columns of P are the n linearly independent eigenvectors of A.

O The statement is false because A must have n distinct eigenvalues for the matrix D and P to exist.

O The statement is true because if A = PDP^-1, then A is diagonalizable by definition.

O The statement is false because the symbol D does not automatically denote a diagonal matrix.

Question

Sophie Le

Mathematics

11 July, 03:20

Let A, P, and D by n x n matrices. Mark each statement true or false. Justify each answer.

a. A is diagonalizable if A = PDP^-1 for some matrix D and some invertible matrix P. Choose the correct answer below.

O The statement is true because the columns of P are the n linearly independent eigenvectors of A.

O The statement is false because A must have n distinct eigenvalues for the matrix D and P to exist.

O The statement is true because if A = PDP^-1, then A is diagonalizable by definition.

O The statement is false because the symbol D does not automatically denote a diagonal matrix.

+3

Answers (1)

Know the Answer?

Gunner Stanton · Answer 1

Correct answer is the 4th answer.

Step-by-step explanation:

By definition, a matrix is called diagonalizable if there exists some invertible matrix P and a diagonal matrix D such that AP = PD, which is equivalent to having A = PDP^-1.

We are not told that D is a diagonal matrix, hence, it could be any matrix. If A=PDP^-1 for some matrix D that is not diagonal, it doesn't imply that A is diagonalizable. Consider the following case: Let us take A a non diagonalizable matrix. Take D = A and P = identity matrix of size n. Then, this equation holds, but A is still not diagonalizable.