Let A, B E Rnxn and let R[x] be the set of all polynomials in variable x with coefficients in R. Definition 1: For any p(x) = ₁0 Cx² € R[x] define the "evaluation of p(x) at A” as i=0 p(A) = k Σ GA i=0 = CkAk + CK-1 + +₁A+ coIn, (here Aº = In). = Definition 2: Two matrices A, B E Rnxn are said to commute if AB = BA. Let A Rnxn. Prove that there exists a polynomial p(x) = R[x] of degree at most n²+1 such that p(A) = Onxn. Hint: Consider {In, A, A², An², An²+1} and use the fact that dim(Rn×n) = n². (
Let A, B E Rnxn and let R[x] be the set of all polynomials in variable x with coefficients in R. Definition 1: For any p(x) = ₁0 Cx² € R[x] define the "evaluation of p(x) at A” as i=0 p(A) = k Σ GA i=0 = CkAk + CK-1 + +₁A+ coIn, (here Aº = In). = Definition 2: Two matrices A, B E Rnxn are said to commute if AB = BA. Let A Rnxn. Prove that there exists a polynomial p(x) = R[x] of degree at most n²+1 such that p(A) = Onxn. Hint: Consider {In, A, A², An², An²+1} and use the fact that dim(Rn×n) = n². (
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
(here Aº := In).
p(A)
:=
k
Σ GA
i=0
=
CkAk + Ck-1Ak-1 + ... + C₁ A + coỈn,
Definition 2: Two matrices A, B € Rnxn are said to commute if AB = BA.
Let A Rnxn Prove that there exists a polynomial p(x) = R[x] of degree at most n²+1 such that p(A) = Onxn.
Hint: Consider {In, A, A², ‚An², An²+¹} and use the fact that dim(R¹×¹) n².
(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce2c460-26f0-4070-86e1-b1876f7380a8%2Fb71cad82-18c5-4d26-889b-b06629437b11%2Fh207jl_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
(here Aº := In).
p(A)
:=
k
Σ GA
i=0
=
CkAk + Ck-1Ak-1 + ... + C₁ A + coỈn,
Definition 2: Two matrices A, B € Rnxn are said to commute if AB = BA.
Let A Rnxn Prove that there exists a polynomial p(x) = R[x] of degree at most n²+1 such that p(A) = Onxn.
Hint: Consider {In, A, A², ‚An², An²+¹} and use the fact that dim(R¹×¹) n².
(
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