Let A = 2 (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. (b) Find a vector v € R² such that (A + I)v = u, and show by direct calculation that P-¹AP = B where P = [u_v] and B = 0 1.
Let A = 2 (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. (b) Find a vector v € R² such that (A + I)v = u, and show by direct calculation that P-¹AP = B where P = [u_v] and B = 0 1.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![Let
A =
(a) By calculating and factorizing C₁(2), the characteristic polynomial of
A, show that -1 is the only eigenvalue of A, and then find a corre-
sponding eigenvector u.
-9 2
(b) Find a vector v R2 such that (A + I)v = u, and show by direct
calculation that P-¹AP = B where
(c) Using the fact that
P = [u v] and B =
Bn
PA
0
= (-1)" [ 7]
for every integer n, find An explicitly in terms of n. Hint: Begin by
using the equality A = PBP-¹ to express A" in terms of P and B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8a9802b-ded9-4a7b-93c8-71218002814a%2Ff41a299c-3003-4c29-828c-ab6ab47917bb%2Faqgj56b_processed.png&w=3840&q=75)
Transcribed Image Text:Let
A =
(a) By calculating and factorizing C₁(2), the characteristic polynomial of
A, show that -1 is the only eigenvalue of A, and then find a corre-
sponding eigenvector u.
-9 2
(b) Find a vector v R2 such that (A + I)v = u, and show by direct
calculation that P-¹AP = B where
(c) Using the fact that
P = [u v] and B =
Bn
PA
0
= (-1)" [ 7]
for every integer n, find An explicitly in terms of n. Hint: Begin by
using the equality A = PBP-¹ to express A" in terms of P and B.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 7 steps with 7 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education