Question

Question: In Exercises 21 and 22, A, B, Pand Dare \(n \times n\) matrices. Mark each statement True or False. Justify each answer.(Study Theorem 5 and 6 and the examples in this section carefully before you try these exercises.)

1. A is diagonalizable if \(A = PD{P^{ - {\bf{1}}}}\) for some matrix D and some invertible matrix P.
2. If \({\mathbb{R}^n}\) has a basis of eigenvectors of A, then A is diagonziable.
3. A is diagonlizable if and only if A has n eigenvalues, counting multiplicities.
4. If A is diagonizable, then A is invertible.

Short answer

a. The given statement is false.

b. The given statement is true.

c. The given statement is false.

d. The given statement is false.

Step-by-step solution

Find the answer for part (a)

If matrix A is diagonalizable, then \(A = PD{P^{ - 1}}\), where matrix D is diagonal, and matrix Pis invertible.

In the statement, it is not mentioned that Dis diagonalizable.

Therefore, the given statement is false.

Find the answer for part (b)

If \({\mathbb{R}^n}\) has a basis of eigenvectors, then matrix A is diagonalizable.

As \({\mathbb{R}^n}\) has a basis of eigenvectors, that there are enough eigenvectors for the basis and the basis \({\mathbb{R}^n}\) is known as eigenvector basis. Thus, A is diagonalizable.

Therefore, the given statement is true.

Find the answer for part (c)

For diagonalization, the matrix must be invertible.

Therefore, the given statement is false.

Find the answer for part (d)

According to the invertible matrix theorem, zero must not be the eigenvalue of an invertible matrix. But for a diagonalizable matrix, zero may or may not be the eigenvalue.

So, A will not be essentially invertible as it is diagonalizable.

Therefore, the given statement is false.