1. For the matrix A=12²5¹ 7 3 6-12-5, (a) Compute the characteristic polynomial f(x) = det(xI - A). (b) Evaluate A² and A³. (c) Using (a) and (b), verify that f(A) = 0. (d) Compute the minimal polynomial m(x) of A, and verify that m(x) divides f(x). (e) The ring R[A] is the set of all polynomials in A with real coefficients, and this is also a vector space over R. What is the dimension of this vector space? Give an explicit basis for R[A]. (f) Show that the ring R[A] has zero divisors (and therefore it is not a field).
1. For the matrix A=12²5¹ 7 3 6-12-5, (a) Compute the characteristic polynomial f(x) = det(xI - A). (b) Evaluate A² and A³. (c) Using (a) and (b), verify that f(A) = 0. (d) Compute the minimal polynomial m(x) of A, and verify that m(x) divides f(x). (e) The ring R[A] is the set of all polynomials in A with real coefficients, and this is also a vector space over R. What is the dimension of this vector space? Give an explicit basis for R[A]. (f) Show that the ring R[A] has zero divisors (and therefore it is not a field).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![1.
For the matrix A =
-2 6 3
-3 7 3
6-12-5
9
(a) Compute the characteristic polynomial f(x) = det(xI - A).
(b) Evaluate A² and A³.
(c) Using (a) and (b), verify that f(A) = 0.
(d) Compute the minimal polynomial m(x) of A, and verify that m(x) divides f(x).
(e) The ring R[A] is the set of all polynomials in A with real coefficients, and this is
also a vector space over R. What is the dimension of this vector space? Give an
explicit basis for R[A].
(f) Show that the ring R[A] has zero divisors (and therefore it is not a field).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F244c6516-c7e6-4ded-b620-989e67bc99ea%2F1a71c016-a037-4ddb-aaf6-99ba0cd24d11%2Feev1wqa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
For the matrix A =
-2 6 3
-3 7 3
6-12-5
9
(a) Compute the characteristic polynomial f(x) = det(xI - A).
(b) Evaluate A² and A³.
(c) Using (a) and (b), verify that f(A) = 0.
(d) Compute the minimal polynomial m(x) of A, and verify that m(x) divides f(x).
(e) The ring R[A] is the set of all polynomials in A with real coefficients, and this is
also a vector space over R. What is the dimension of this vector space? Give an
explicit basis for R[A].
(f) Show that the ring R[A] has zero divisors (and therefore it is not a field).
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