Let M₂ (R) be the set of 2 x 2 matrices with real number entries, i.e., b M₂ (R) = { [ a &] | a,b,c,d ER}. Define matrix addition and matrix multiplication in M₂ (R) as follows b₁ + b₂ a₁ b₁ c₁d₁ a2 b₂ 0₂ d₂ = [3 a₁ + a₂ C₁+0₂ 1₁ d₁ + d₂ [a ;] + [ ][ a2 b₂ a₁a₂+ b₁c₂ a1b₂ + b₁d₂ a1 b₁ c₁d₁ 0₂ d₂ = 1. Show that M₂ (R) is a ring under addition and multiplication defined above. [No need to show matrix addition and matrix multiplication are binary operations in M₂(R).] 2. Is the ring M₂ (R) commutative? Justify. 3. Does the ring have a unity? If so, what is it?
Let M₂ (R) be the set of 2 x 2 matrices with real number entries, i.e., b M₂ (R) = { [ a &] | a,b,c,d ER}. Define matrix addition and matrix multiplication in M₂ (R) as follows b₁ + b₂ a₁ b₁ c₁d₁ a2 b₂ 0₂ d₂ = [3 a₁ + a₂ C₁+0₂ 1₁ d₁ + d₂ [a ;] + [ ][ a2 b₂ a₁a₂+ b₁c₂ a1b₂ + b₁d₂ a1 b₁ c₁d₁ 0₂ d₂ = 1. Show that M₂ (R) is a ring under addition and multiplication defined above. [No need to show matrix addition and matrix multiplication are binary operations in M₂(R).] 2. Is the ring M₂ (R) commutative? Justify. 3. Does the ring have a unity? If so, what is it?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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![Let M₂ (R) be the set of 2 x 2 matrices with real number entries, i.e.,
a b
M₂ (R) = {[%
• { [ @ $ ] 1a,b,c,dER}.
d
Define matrix addition and matrix multiplication in M₂ (R) as follows
a1 b₁
d₁
a₁ + a₂ b₁ + b₂
;] + [
=
az b₂
0₂ d₂
[
C1
a1
1][² :]
a2 b₂
d₂
a₁a2+ b₁c₂ a1b₂ + b₁d₂
c₁a₂+d₂c₂ c₁b₂+d₁d₂
:].
c₁d₁
1. Show that M₂ (R) is a ring under addition and multiplication defined above.
[No need to show matrix addition and matrix multiplication are binary operations in M₂ (R).]
2. Is the ring M₂ (R) commutative? Justify.
3. Does the ring have a unity? If so, what is it?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1c653e3-7876-48cf-8145-f982de6a0d95%2F9e68e1e5-2985-4f5c-81a7-cd8054a3dc2f%2Fxxhn846_processed.png&w=3840&q=75)
Transcribed Image Text:Let M₂ (R) be the set of 2 x 2 matrices with real number entries, i.e.,
a b
M₂ (R) = {[%
• { [ @ $ ] 1a,b,c,dER}.
d
Define matrix addition and matrix multiplication in M₂ (R) as follows
a1 b₁
d₁
a₁ + a₂ b₁ + b₂
;] + [
=
az b₂
0₂ d₂
[
C1
a1
1][² :]
a2 b₂
d₂
a₁a2+ b₁c₂ a1b₂ + b₁d₂
c₁a₂+d₂c₂ c₁b₂+d₁d₂
:].
c₁d₁
1. Show that M₂ (R) is a ring under addition and multiplication defined above.
[No need to show matrix addition and matrix multiplication are binary operations in M₂ (R).]
2. Is the ring M₂ (R) commutative? Justify.
3. Does the ring have a unity? If so, what is it?
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