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).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 16EQ
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