(a) For the parts (i)-(ii), let A, B E Rnxn be such that A and B commute. Furthermore, let p(x), q(x) E R[x]. (i) Prove that p(A) and B commute. (ii) Is it true that p(A) and q(B) commute? Justify your answer.
(a) For the parts (i)-(ii), let A, B E Rnxn be such that A and B commute. Furthermore, let p(x), q(x) E R[x]. (i) Prove that p(A) and B commute. (ii) Is it true that p(A) and q(B) commute? Justify your answer.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 65E
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![Let A, B € Rn×n and let R[x] be the set of all polynomials in variable x with coefficients in R.
Definition 1: For any p(x) = Σo C₁x¹ = R[x] define the “evaluation of p(x) at A" as
k
i=0
p(A)
:=
k
Σ GA
i=0
=
CkAk + Ck-1Ak-1 + ... + C₁ A + coỈn, (here Aº := In).
Definition 2: Two matrices A, B € Rnxn are said to commute if AB = BA.
ιχη
(a) For the parts (i)-(ii), let A, B € Rnxn be such that A and B commute. Furthermore, let p(x), q(x) = R[x].
(i) Prove that p(A) and B commute.
(ii) Is it true that p(A) and q(B) commute? Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce2c460-26f0-4070-86e1-b1876f7380a8%2F007b24ed-7093-446c-a88e-1cfcc884068b%2Fyhd89ny_processed.png&w=3840&q=75)
Transcribed Image Text:Let A, B € Rn×n and let R[x] be the set of all polynomials in variable x with coefficients in R.
Definition 1: For any p(x) = Σo C₁x¹ = R[x] define the “evaluation of p(x) at A" as
k
i=0
p(A)
:=
k
Σ GA
i=0
=
CkAk + Ck-1Ak-1 + ... + C₁ A + coỈn, (here Aº := In).
Definition 2: Two matrices A, B € Rnxn are said to commute if AB = BA.
ιχη
(a) For the parts (i)-(ii), let A, B € Rnxn be such that A and B commute. Furthermore, let p(x), q(x) = R[x].
(i) Prove that p(A) and B commute.
(ii) Is it true that p(A) and q(B) commute? Justify your answer.
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