For a natural number n, let C, = {(a¡j) E M„(R): aij = aki for all i, j,k,l such that j– i=,l- k}. (a) Exactly one of the two matrices below is an element of C3. Which one? (0 1 2 Aj = |1 2 0 (2 0 1 (0 1 2 Az = |2 0 1 1 2 0, (b) Let A be the matrix you chose in part (a). Compute A². Check that A? € C3, and write out a brief explanation (one or two sentences is enough) explaining how you checked. (c) Give a complete proof that C3 is a ring. You may assume that M3(R) is a ring. Hint: make as much use of that assumption as possible in your proof.
For a natural number n, let C, = {(a¡j) E M„(R): aij = aki for all i, j,k,l such that j– i=,l- k}. (a) Exactly one of the two matrices below is an element of C3. Which one? (0 1 2 Aj = |1 2 0 (2 0 1 (0 1 2 Az = |2 0 1 1 2 0, (b) Let A be the matrix you chose in part (a). Compute A². Check that A? € C3, and write out a brief explanation (one or two sentences is enough) explaining how you checked. (c) Give a complete proof that C3 is a ring. You may assume that M3(R) is a ring. Hint: make as much use of that assumption as possible in your proof.
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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Question
![For a natural number n, let
C, = {(aij) E M„(R) : ai :
= aki for all i, j,k,l such that j–i=,1– k}.
(a) Exactly one of the two matrices below is an element of C3. Which one?
(0 1 2
A1 = |1 2 0
20 1
(0 1 2)
A2 = |2 0 1
1 2 0
(b) Let A be the matrix you chose in part (a). Compute A?. Check that A? e C3, and
write out a brief explanation (one or two sentences is enough) explaining how
you checked.
(c) Give a complete proof that C3 is a ring. You may assume that M3(R) is a ring.
Hint: make as much use of that assumption as possible in your proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F283b3a0d-0821-472c-a1ce-fcf162c3f2c6%2F47b92e2b-ea9e-4124-9c28-692cbefad939%2F8x2ah1_processed.png&w=3840&q=75)
Transcribed Image Text:For a natural number n, let
C, = {(aij) E M„(R) : ai :
= aki for all i, j,k,l such that j–i=,1– k}.
(a) Exactly one of the two matrices below is an element of C3. Which one?
(0 1 2
A1 = |1 2 0
20 1
(0 1 2)
A2 = |2 0 1
1 2 0
(b) Let A be the matrix you chose in part (a). Compute A?. Check that A? e C3, and
write out a brief explanation (one or two sentences is enough) explaining how
you checked.
(c) Give a complete proof that C3 is a ring. You may assume that M3(R) is a ring.
Hint: make as much use of that assumption as possible in your proof.
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