Problem 1. ? ? ? ? ? Are the following statements true or false for a square matrix A? 1. A number c is an eigenvalue of A if and only if the equation (A − cI)x= 0 has a nontrivial solution X. 2. A matrix A is singular if and only if 0 is an eigenvalue of A. 3. To find the eigenvalues of A, reduce A to echelon form. 4. If Ax = 2x for some vector X and some scalar λ, then is an eigenvalue of A. 5. An n x n matrix A is diagonalizable if A has n linearly independent eigenvectors.
Problem 1. ? ? ? ? ? Are the following statements true or false for a square matrix A? 1. A number c is an eigenvalue of A if and only if the equation (A − cI)x= 0 has a nontrivial solution X. 2. A matrix A is singular if and only if 0 is an eigenvalue of A. 3. To find the eigenvalues of A, reduce A to echelon form. 4. If Ax = 2x for some vector X and some scalar λ, then is an eigenvalue of A. 5. An n x n matrix A is diagonalizable if A has 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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Transcribed Image Text:Problem 1.
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Are the following statements true or false for a square matrix A?
1. A number c is an eigenvalue of A if and only if the equation (A - cI)x = 0 has a nontrivial solution X.
2. A matrix A is singular if and only if 0 is an eigenvalue of A.
3. To find the eigenvalues of A, reduce A to echelon form.
4. If Ax = 2x for some vector x and some scalar λ, then λ is an eigenvalue of A.
5. An n x n matrix A is diagonalizable if A has ʼn linearly independent eigenvectors.
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