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Which of the following statements are true? Select all that apply. JA. IFA is invertible and 1 is an eigenvalue ofA, then 1 is also an eigenvalue of A-1. OB. IHA contains a row or column of zeros, then O is an eigenvalue of A c. The eigenvalues of an upper triangular matrix A are exactiy the nonzero entries on the diagonal of A.

Which of the following statements are true? Select all that apply. JA. IFA is invertible and 1 is an eigenvalue ofA, then 1 is also an eigenvalue of A-1. OB. IHA contains a row or column of zeros, then O is an eigenvalue of A c. The eigenvalues of an upper triangular matrix A are exactiy the nonzero entries on the diagonal of A.

Advanced Engineering Mathematics
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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Determine which statements about square matrices in parts (a)-(f) below are true. a. Which of the following statements are true? Select all that apply. A. IfA is invertible and 1 is an eigenvalue of A, then 1 is also an eigenvalue of A-1. B. IfA contains a row or column of zeros, then 0 is an eigenvalue of A. ] c. The eigenvalues of an upper triangular matrix A are exactly the nonzero entries on the diagonal of A. b. Which of the following statements are true? Select all that apply. |A. The matrices A and AT have the same eigenvalues, counting multiplicities. B. The sum of two eigenvectors of a matrix A is also an eigenvector of A. C. Each eigenvalue of A is also an eigenvalue of A?. D. Each eigenvector of A is also an eigenvector of A?. E. Each eigenvector of an invertible matrix A is also an eigenvector of A 1. c. Which of the following statements are true? Select all that apply. A. Eigenvalues must be nonzero scalars. B. Eigenvectors must be nonzero vectors. c. Two eigenvectors corresponding to the same eigenvalue are always linearly dependent. D. A nonzero vector cannot correspond to two different eigenvalues of A. d. Which of the following statements are true? Select all that apply. A. Similar matrices always have exactly the same eigenvectors. B. IfA and B are invertible nxn matrices, then AB is similar to BA. c. IfA is similar to a diagonalizable matrix B, then A is also diagonalizable. D. Similar matrices always have exactly the same eigenvalues.
Transcribed Image Text:Determine which statements about square matrices in parts (a)-(f) below are true. a. Which of the following statements are true? Select all that apply. A. IfA is invertible and 1 is an eigenvalue of A, then 1 is also an eigenvalue of A-1. B. IfA contains a row or column of zeros, then 0 is an eigenvalue of A. ] c. The eigenvalues of an upper triangular matrix A are exactly the nonzero entries on the diagonal of A. b. Which of the following statements are true? Select all that apply. |A. The matrices A and AT have the same eigenvalues, counting multiplicities. B. The sum of two eigenvectors of a matrix A is also an eigenvector of A. C. Each eigenvalue of A is also an eigenvalue of A?. D. Each eigenvector of A is also an eigenvector of A?. E. Each eigenvector of an invertible matrix A is also an eigenvector of A 1. c. Which of the following statements are true? Select all that apply. A. Eigenvalues must be nonzero scalars. B. Eigenvectors must be nonzero vectors. c. Two eigenvectors corresponding to the same eigenvalue are always linearly dependent. D. A nonzero vector cannot correspond to two different eigenvalues of A. d. Which of the following statements are true? Select all that apply. A. Similar matrices always have exactly the same eigenvectors. B. IfA and B are invertible nxn matrices, then AB is similar to BA. c. IfA is similar to a diagonalizable matrix B, then A is also diagonalizable. D. Similar matrices always have exactly the same eigenvalues.
e. Which of the following statements are true? Select all that apply. A. IfA is row equivalent to the identity matrix I, then A is diagonalizable. B. IfA is diagonalizable, then the columns of A are linearly independent. C. IfA is an nxn diagonalizable matrix, then each vector in R" can be written as a linear combination of eigenvectors of A. D. If a 5x5 matrix A has fewer than 5 distinct eigenvalues, then A is not diagonalizable. | E. A (square) matrix A is invertible if and only if there is a coordinate system in which the transformation x--Ax is represented by a diagonal matrix. f. Which of the following statements are true? Select all that apply. A. There exists a 2 x2 matrix that has no eigenvectors in R2. B. If each vector e, in the standard basis for R" is an eigenvector of A, then A is a diagonal matrix. C. An nxn matrix with n linearly independent eigenvectors is invertible.
Transcribed Image Text:e. Which of the following statements are true? Select all that apply. A. IfA is row equivalent to the identity matrix I, then A is diagonalizable. B. IfA is diagonalizable, then the columns of A are linearly independent. C. IfA is an nxn diagonalizable matrix, then each vector in R" can be written as a linear combination of eigenvectors of A. D. If a 5x5 matrix A has fewer than 5 distinct eigenvalues, then A is not diagonalizable. | E. A (square) matrix A is invertible if and only if there is a coordinate system in which the transformation x--Ax is represented by a diagonal matrix. f. Which of the following statements are true? Select all that apply. A. There exists a 2 x2 matrix that has no eigenvectors in R2. B. If each vector e, in the standard basis for R" is an eigenvector of A, then A is a diagonal matrix. C. An nxn matrix with n linearly independent eigenvectors is invertible.
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