36. Prove that if λ is an eigenvalue of A and x is a corresponding eigenvector, then sλ is an eigenvalue of sA for every scalar s, and x is a corresponding eigenvector

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 29EQ
icon
Related questions
Question
35. TRUE/FALSE. Justification is needed.
a. If 0 is an eigenvalue of a matrix A, then A² is singular
b.
The eigenvalues of a matrix A are the same as the eigenvalues of the reduced row
echelon form of A
c.
If 0 is an eigenvalue of a matrix A, then the set of columns of A is linearly independent
d. If the characteristic polynomial of a matrix A is p(x) = 1² + 1, then A is invertible
36. Prove that if λ is an eigenvalue of A and x is a corresponding eigenvector, then så is an
eigenvalue of sA for every scalar s, and x is a corresponding eigenvector
3
-1 0
37. Using diagonalization, find An if n is a positive integer and A= -1
2
0
-1
-1
3
Transcribed Image Text:35. TRUE/FALSE. Justification is needed. a. If 0 is an eigenvalue of a matrix A, then A² is singular b. The eigenvalues of a matrix A are the same as the eigenvalues of the reduced row echelon form of A c. If 0 is an eigenvalue of a matrix A, then the set of columns of A is linearly independent d. If the characteristic polynomial of a matrix A is p(x) = 1² + 1, then A is invertible 36. Prove that if λ is an eigenvalue of A and x is a corresponding eigenvector, then så is an eigenvalue of sA for every scalar s, and x is a corresponding eigenvector 3 -1 0 37. Using diagonalization, find An if n is a positive integer and A= -1 2 0 -1 -1 3
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage