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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