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)
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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,...
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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.
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.
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