Assume A, B, P, and D are nxn matrices. Determine whether the folowing statements are true or false. Justify each answer.a. A matrix A is diagonalizable if A has n eigenvectors.A. The statement is true. A diagonalizable matrix must have a minimum of n linearly independent eigenvectors.B. The statement is false. A matrix is diagonalizable if and only if it has n- 1 linearly independent eigenvectors.c. The statement is true. A diagonalizable matrix must have more than one linearly independent eigenvector.D. The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.b. If A is diagonalizable, then A has n distinct eigenvalues.A. The statement is false. A diagonalizable matrix must have more than n eigenvalues.B. The statement is true. A diagonalizable matrix must have exactly n eigenvalues.c. The statement is true. A diagonalizable matrix must have n distinct eigenvalues.D. The statement is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent eigenvectors.c. If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.A. The statement is true. AP = PD implies that the columns of the product PD are eigenvalues that correspond to the eigenvectors of A.B. The statement is false. AP = PD cannot imply that A is diagonalizable, so the columns of P may not be eigenvectors of A.c. The statement is true. Let v be a nonzero column in P and let 2 be the corresponding diagonal element in D. Then AP = PD implies that Av =iv, which means that v is an eigenvector of A.D. The statement is false. If P has a zero column, then it is not linearly independent and so A is not diagonalizable.d. If A is invertible, then A is diagonalizable.A. The statement is false. Invertible matrices always have a maximum of n linearly independent eigenvectors, making it not diagonalizable.B. The statement is false. An invertible matrix may have fewer than n linearly independent eigenvectors, making it not diagonalizable.C. The statement is true. If a matrix is invertible, then it has n linearly independent eigenvectors, making it diagonalizable.D. The statement is true. A diagonalizable matrix is invertible, so an invertible matrix is diagonalizable.

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Assume A, B, P, and D are nxn matrices. Determine whether the folowing statements are true or false. Justify each answer. a. A matrix A is diagonalizable if A has n eigenvectors. A. The statement is true. A diagonalizable matrix must have a minimum of n linearly independent eigenvectors. B. The statement is false. A matrix is diagonalizable if and only if it has n-1 linearly independent eigenvectors. C. The statement is true. A diagonalizable matrix must have more than one linearly independent eigenvector. D. The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.

Assume A, B, P, and D are nxn matrices. Determine whether the folowing statements are true or false. Justify each answer. a. A matrix A is diagonalizable if A has n eigenvectors. A. The statement is true. A diagonalizable matrix must have a minimum of n linearly independent eigenvectors. B. The statement is false. A matrix is diagonalizable if and only if it has n-1 linearly independent eigenvectors. C. The statement is true. A diagonalizable matrix must have more than one linearly independent eigenvector. D. The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Assume A, B, P, and D are nxn matrices. Determine whether the folowing statements are true or false. Justify each answer.
a. A matrix A is diagonalizable if A has n eigenvectors.
A. The statement is true. A diagonalizable matrix must have a minimum of n linearly independent eigenvectors.
B. The statement is false. A matrix is diagonalizable if and only if it has n- 1 linearly independent eigenvectors.
c. The statement is true. A diagonalizable matrix must have more than one linearly independent eigenvector.
D. The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.
b. If A is diagonalizable, then A has n distinct eigenvalues.
A. The statement is false. A diagonalizable matrix must have more than n eigenvalues.
B. The statement is true. A diagonalizable matrix must have exactly n eigenvalues.
c. The statement is true. A diagonalizable matrix must have n distinct eigenvalues.
D. The statement is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent eigenvectors.
c. If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.
A. The statement is true. AP = PD implies that the columns of the product PD are eigenvalues that correspond to the eigenvectors of A.
B. The statement is false. AP = PD cannot imply that A is diagonalizable, so the columns of P may not be eigenvectors of A.
c. The statement is true. Let v be a nonzero column in P and let 2 be the corresponding diagonal element in D. Then AP = PD implies that Av =iv, which means that v is an eigenvector of A.
D. The statement is false. If P has a zero column, then it is not linearly independent and so A is not diagonalizable.
d. If A is invertible, then A is diagonalizable.
A. The statement is false. Invertible matrices always have a maximum of n linearly independent eigenvectors, making it not diagonalizable.
B. The statement is false. An invertible matrix may have fewer than n linearly independent eigenvectors, making it not diagonalizable.
C. The statement is true. If a matrix is invertible, then it has n linearly independent eigenvectors, making it diagonalizable.
D. The statement is true. A diagonalizable matrix is invertible, so an invertible matrix is diagonalizable.

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