-8 k A -4 has two distinct real eigenvalues if and only if k < * 00 II

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 24EQ
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-8 k
A
-4
has two distinct real eigenvalues if and only if k <
* 00
Transcribed Image Text:-8 k A -4 has two distinct real eigenvalues if and only if k < * 00
A is an n X n matrix.
Check the true statements below:
A. Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy.
B. A matrix A is not invertible if and only if 0 is an eigenvalue of A.
C. A number c is an eigenvalue of A if and only if the equation (A – cI)x = 0 has a nontrivial solution x.
D. If Ax = Ax for some vector x, then 1 is an eigenvalue of A.
E. To find the eigenvalues of A, reduce A to echelon form.
Transcribed Image Text:A is an n X n matrix. Check the true statements below: A. Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy. B. A matrix A is not invertible if and only if 0 is an eigenvalue of A. C. A number c is an eigenvalue of A if and only if the equation (A – cI)x = 0 has a nontrivial solution x. D. If Ax = Ax for some vector x, then 1 is an eigenvalue of A. E. To find the eigenvalues of A, reduce A to echelon form.
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