Let A, B € Mnxn (C) be the space of square matrices with complex entries and let O € Mnxn (C) be the zero matrix. In the following, remember that a proof is a justification for all n, not just a few specific examples or values of n. (a) Prove that if A is similar to B (so there is an invertible Q such that A = Q¯¹BQ), then det (A) = det (B). (b) Suppose there is a positive integer k such that Ak : = O. Prove that A is not invertible. (c) Prove that if A² = -A and n is odd, then A is not invertible.

Let A, B € Mnxn (C) be the space of square matrices with complex entries and let O € Mnxn (C) be the zero matrix. In the following, remember that a proof is a justification for all n, not just a few specific examples or values of n. (a) Prove that if A is similar to B (so there is an invertible Q such that A = Q¯¹BQ), then det (A) = det (B). (b) Suppose there is a positive integer k such that Ak : = O. Prove that A is not invertible. (c) Prove that if A² = -A and n is odd, then A is not invertible.

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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