Consider the ring M3(R) of 3 x 3 real matrices. As usual, we denote by 03 the zero-matrix and by I3 the identity matrix. For a matrix A E M3(R) and a polynomial P(X) = anX" + an-1X"-1+...+ a1X + ao E R[X] %3D with ao,..., an ER, we define the following matrix: P(A) := an A" + an-1A"-1 +..+ a1A+ aola E M3(R), and we define the following set: IA := {PER[X] such that P(A) 03}. This set is an ideal of R[X]. (You do not have to prove this fact. A similar result was seen in the course.) In this problem, we study the following matrix: 70 1 0 B:=0 0 1 e M3(R). 0 0 0 (a) Show that the polynomial X e R[X] is an element of the ideal IB. (b) Show that there exists a unique non-zero polynomial PB E R[X] with leading coefficient 1 such that I (PB). This polynomial PB is called the minimal polynomial of the matrix B.
Consider the ring M3(R) of 3 x 3 real matrices. As usual, we denote by 03 the zero-matrix and by I3 the identity matrix. For a matrix A E M3(R) and a polynomial P(X) = anX" + an-1X"-1+...+ a1X + ao E R[X] %3D with ao,..., an ER, we define the following matrix: P(A) := an A" + an-1A"-1 +..+ a1A+ aola E M3(R), and we define the following set: IA := {PER[X] such that P(A) 03}. This set is an ideal of R[X]. (You do not have to prove this fact. A similar result was seen in the course.) In this problem, we study the following matrix: 70 1 0 B:=0 0 1 e M3(R). 0 0 0 (a) Show that the polynomial X e R[X] is an element of the ideal IB. (b) Show that there exists a unique non-zero polynomial PB E R[X] with leading coefficient 1 such that I (PB). This polynomial PB is called the minimal polynomial of the matrix B.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Problem 3. Consider the ring M3(R) of 3 x 3 real matrices. As usual, we denote by 03 the
zero-matrix and by I; the identity matrix.
For a matrix A e M3(R) and a polynomial
P(X) = anX" + an-1X"-1+...+ a1X + ao E R[X]
%3D
with ao, ..., a, E R, we define the following matrix:
P(A) := an A" + an-1A"-1 + ... + a1A + aola E M3(R),
and we define the following set:
IA := {P € R[X] such that P(A) 03}.
This set is an ideal of R[X]. (You do not have to prove this fact. A similar result
was seen in the course.)
In this problem, we study the following matrix:
10 1 0
B:= 0 0 1 E M3(R).
0 0 0
(a) Show that the polynomial X E R[X] is an element of the ideal Ig.
(b) Show that there exists a unique non-zero polynomial PB E R[X] with leading
coefficient 1 such that I (PB). This polynomial PB is called the minimal
polynomial of the matrix B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7068779e-a624-45ba-8c15-d1ec8879533a%2Ff98b415d-94c4-41b7-8599-2bee901e4bd5%2Fsvksmo8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 3. Consider the ring M3(R) of 3 x 3 real matrices. As usual, we denote by 03 the
zero-matrix and by I; the identity matrix.
For a matrix A e M3(R) and a polynomial
P(X) = anX" + an-1X"-1+...+ a1X + ao E R[X]
%3D
with ao, ..., a, E R, we define the following matrix:
P(A) := an A" + an-1A"-1 + ... + a1A + aola E M3(R),
and we define the following set:
IA := {P € R[X] such that P(A) 03}.
This set is an ideal of R[X]. (You do not have to prove this fact. A similar result
was seen in the course.)
In this problem, we study the following matrix:
10 1 0
B:= 0 0 1 E M3(R).
0 0 0
(a) Show that the polynomial X E R[X] is an element of the ideal Ig.
(b) Show that there exists a unique non-zero polynomial PB E R[X] with leading
coefficient 1 such that I (PB). This polynomial PB is called the minimal
polynomial of the matrix B.
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