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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.4: Similarity And Diagonalization
Problem 41EQ:
In general, it is difficult to show that two matrices are similar. However, if two similar matrices...
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