Let M2(R) be the set of 2 x 2 matrices with real number entries, i.e., - {[: :] 1ah ader}. b M2(R) = la, b, c, d e Define matrix addition and matrix multiplication in M2(R) as follows b1 di b2 d2 a1 + a2 b1 + b2 Ci + c2 d1 + d2 a1 a2 C1 C2 b1 di b2 d2 a1a2 + bịC2 a¡b2 + bịd2 Cia2 + dịc2 cıb2 + d1d2 a1 a2 C1 C2 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).]
Let M2(R) be the set of 2 x 2 matrices with real number entries, i.e., - {[: :] 1ah ader}. b M2(R) = la, b, c, d e Define matrix addition and matrix multiplication in M2(R) as follows b1 di b2 d2 a1 + a2 b1 + b2 Ci + c2 d1 + d2 a1 a2 C1 C2 b1 di b2 d2 a1a2 + bịC2 a¡b2 + bịd2 Cia2 + dịc2 cıb2 + d1d2 a1 a2 C1 C2 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).]
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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![Let M2 (R) be the set of 2 x 2 matrices with real number entries, i.e.,
{[: :] nmader}.
| |a, b, c, d € R
Define matrix addition and matrix multiplication in M2(R) as follows
a1 b1
+
b2
d2
a1 + a2 b1 + b2
С1 + с2 d + dz
C1
di
C2
a1 b1
di
a1a2 + b1c2
C1a2 + d1c2 cıb2+d1d2
a,b2 + bid2
b2
d2
a2
C1
C2
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?
4. Is M2(R) a field? Justify.
{{
||x, y, z € R> is a subring of M2(R).
5. Show that T =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F238e88b5-359a-4c1f-8db2-70276d834298%2F630e6ba4-e6fc-460b-abd2-f528a8dac10b%2Fvqmdmag_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let M2 (R) be the set of 2 x 2 matrices with real number entries, i.e.,
{[: :] nmader}.
| |a, b, c, d € R
Define matrix addition and matrix multiplication in M2(R) as follows
a1 b1
+
b2
d2
a1 + a2 b1 + b2
С1 + с2 d + dz
C1
di
C2
a1 b1
di
a1a2 + b1c2
C1a2 + d1c2 cıb2+d1d2
a,b2 + bid2
b2
d2
a2
C1
C2
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?
4. Is M2(R) a field? Justify.
{{
||x, y, z € R> is a subring of M2(R).
5. Show that T =
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