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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)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.8: Determinants
Problem 37E
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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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