(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 x 3 matrix а 1 Ma := a 1 а Show that p(a) p'(a) p"(a) p(a) p'(a) p(a). p(Ma) (Hint: prove it first for monomials of the form x" by induction on n, and use the principle of linearity to prove this for all polynomials.) (b) Use this to determine the minimal polynomial of Ma.
(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 x 3 matrix а 1 Ma := a 1 а Show that p(a) p'(a) p"(a) p(a) p'(a) p(a). p(Ma) (Hint: prove it first for monomials of the form x" by induction on n, and use the principle of linearity to prove this for all polynomials.) (b) Use this to determine the minimal polynomial of Ma.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.3: Factorization In F [x]
Problem 4E: Write each of the following polynomials as a products of its leading coefficient and a finite number...
Related questions
Question
![(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3
x 3 matrix
a
1
Ma
a
1
a
Show that
p(a) p'(a) p"(a)
p(Ma):
p(a)
p'(a)
p(a).
(Hint: prove it first for monomials of the form x" by induction on n, and use the principle of
linearity to prove this for all polynomials.)
(b) Use this to determine the minimal polynomial of Ma.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a2b9261-2a0d-4e40-836c-2d268431eab4%2F4ecdbc20-2ec7-4229-b023-cb769a3defe0%2Fqvbasm9_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3
x 3 matrix
a
1
Ma
a
1
a
Show that
p(a) p'(a) p"(a)
p(Ma):
p(a)
p'(a)
p(a).
(Hint: prove it first for monomials of the form x" by induction on n, and use the principle of
linearity to prove this for all polynomials.)
(b) Use this to determine the minimal polynomial of Ma.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage