P7.2. Let U3(Z) be the subset of all upper triangular (3 × 3)-matrices with integer entries; €Z}. U₂(Z) := {[ n(amxm + am-1 A1 A₂ A3 0 a4 a5 A₁, A2,..., α6 €. 0 Ο ασ. (i) Verify that U3(Z) is a subring of the ring of all (3×3)-matrices with integer entries. (ii) Given the matrix 010 N: 0 0 1 000 2 let ŋ: Z[x] →U3(Z) xm-1 + + a₁ x + ao) = am Nm + am-1 Find a polynomial g in Z[x] such that Ker(n) = (g). be the ring homomorphism defined by Nm-1 + + a₁ N + a₁ I.
P7.2. Let U3(Z) be the subset of all upper triangular (3 × 3)-matrices with integer entries; €Z}. U₂(Z) := {[ n(amxm + am-1 A1 A₂ A3 0 a4 a5 A₁, A2,..., α6 €. 0 Ο ασ. (i) Verify that U3(Z) is a subring of the ring of all (3×3)-matrices with integer entries. (ii) Given the matrix 010 N: 0 0 1 000 2 let ŋ: Z[x] →U3(Z) xm-1 + + a₁ x + ao) = am Nm + am-1 Find a polynomial g in Z[x] such that Ker(n) = (g). be the ring homomorphism defined by Nm-1 + + a₁ N + a₁ I.
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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![P7.2. Let U3(Z) be the subset of all upper triangular (3 × 3)-matrices with integer entries;
ª6 EZ}.
U3(Z) :=
A1 A₂ A3
Ο α, ας
0 0 ασ]
{[8
A1, A₂,..., α6
(i) Verify that U3(Z) is a subring of the ring of all (3×3)-matrices with integer entries.
(ii) Given the matrix
0 1 0
N:= 0
000
9
let ŋ: Z[x] →U3(Z) be the ring homomorphism defined by
n(amxm + am-1xm−1+ + a₁x + ao) = am Nm + am_1 Nm−¹ +
Find a polynomial g in Z[x] such that Ker(n) = (g).
+ a₁ N + a。I.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e9b89df-c7d2-4fa7-b4a5-32520659af16%2F716e5c66-dbd6-4cc3-b0d4-01ae80c2a8b3%2F4cjxlq_processed.png&w=3840&q=75)
Transcribed Image Text:P7.2. Let U3(Z) be the subset of all upper triangular (3 × 3)-matrices with integer entries;
ª6 EZ}.
U3(Z) :=
A1 A₂ A3
Ο α, ας
0 0 ασ]
{[8
A1, A₂,..., α6
(i) Verify that U3(Z) is a subring of the ring of all (3×3)-matrices with integer entries.
(ii) Given the matrix
0 1 0
N:= 0
000
9
let ŋ: Z[x] →U3(Z) be the ring homomorphism defined by
n(amxm + am-1xm−1+ + a₁x + ao) = am Nm + am_1 Nm−¹ +
Find a polynomial g in Z[x] such that Ker(n) = (g).
+ a₁ N + a。I.
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