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Which of the following statements hold for the minimal polynomial of a matrix B. (i) The minimal polynomial of a matrix B divides the characteristic polynomial of the matrix B. (ii) The minimal polynomial of a matrix B is a monic polynomial of the smallest degree in the set {f(1) f(B)=0}. (iii) The minimal polynomial of a matrix B divides every polynomial which annihilates matrix B. A. Only (i) and (ii) B. None of the given statements. C. Only (i) and (iii) D. All of the given statements. E. Only (ii) and (iii)

Which of the following statements hold for the minimal polynomial of a matrix B. (i) The minimal polynomial of a matrix B divides the characteristic polynomial of the matrix B. (ii) The minimal polynomial of a matrix B is a monic polynomial of the smallest degree in the set {f(1) f(B)=0}. (iii) The minimal polynomial of a matrix B divides every polynomial which annihilates matrix B. A. Only (i) and (ii) B. None of the given statements. C. Only (i) and (iii) D. All of the given statements. E. Only (ii) and (iii)

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
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Which of the following statements hold for the minimal polynomial of a matrix B. (i) The minimal polynomial of a matrix B divides the characteristic polynomial of the matrix B. (ii) The minimal polynomial of a matrix B is a monic polynomial of the smallest degree in the set {f(2) f(B)=0}. (iii) The minimal polynomial of a matrix B divides every polynomial which annihilates matrix B. A. Only (i) and (ii) B. None of the given statements. C. Only (i) and (iii) D. All of the given statements. E. Only (ii) and (iii)
Transcribed Image Text:Which of the following statements hold for the minimal polynomial of a matrix B. (i) The minimal polynomial of a matrix B divides the characteristic polynomial of the matrix B. (ii) The minimal polynomial of a matrix B is a monic polynomial of the smallest degree in the set {f(2) f(B)=0}. (iii) The minimal polynomial of a matrix B divides every polynomial which annihilates matrix B. A. Only (i) and (ii) B. None of the given statements. C. Only (i) and (iii) D. All of the given statements. E. Only (ii) and (iii)
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