Let B be an n x n matrix over a field F, and suppose that B has minimal polynomialx2 + x + 1.(i) Show that B is invertible.(ii)(iii)(iv)Prove that the minimal polynomial of B-1 is also x? + x + 1.Prove that ifF = C and n is odd, then B is not similar to B-1.Prove that if F = R and n is even, then B is similar to B-1.

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Let B be an n × n matrix over a field F, and suppose that B has minimal polynomial x2 +x +1. (i) Show that B is invertible. (ii) Prove that the minimal polynomial of B-1 is also x? + x +1. (iii) Prove that if F = C and n is odd, then B is not similar to B-1. (iv) Prove that if F = R and n is even, then B is similar to B-1.

Let B be an n × n matrix over a field F, and suppose that B has minimal polynomial x2 +x +1. (i) Show that B is invertible. (ii) Prove that the minimal polynomial of B-1 is also x? + x +1. (iii) Prove that if F = C and n is odd, then B is not similar to B-1. (iv) Prove that if F = R and n is even, then B is similar to B-1.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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