Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
Let B be an n x n matrix over a field F, and suppose that B has minimal polynomial x2 + 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.
Transcribed Image Text:Let B be an n x n matrix over a field F, and suppose that B has minimal polynomial x2 + 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.
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Linear Transformation
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS