6. Let B Rnxn be an arbitrary matrix. Our goal is to prove that there exists a positive integer k such that Rn = R(Bk) + N(B¹). (a) Prove that if B is a nonsingular matrix, then (3) holds with k = 1. Next we want to show that (3) holds for some k even if B is a singular matrix. (b) Prove that N(B²) ≤ N(B²+¹) and R(B²) 2 R(B²+¹), for l = 0, 1, 2, .... Here Bº := I. (c) Prove that there exists an integer j such that N(B¹) = N(Bi+¹). (3)

6. Let B Rnxn be an arbitrary matrix. Our goal is to prove that there exists a positive integer k such that Rn = R(Bk) + N(B¹). (a) Prove that if B is a nonsingular matrix, then (3) holds with k = 1. Next we want to show that (3) holds for some k even if B is a singular matrix. (b) Prove that N(B²) ≤ N(B²+¹) and R(B²) 2 R(B²+¹), for l = 0, 1, 2, .... Here Bº := I. (c) Prove that there exists an integer j such that N(B¹) = N(Bi+¹). (3)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Matrices

Section2.1: Operations With Matrices

Problem 77E

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