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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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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