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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
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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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.
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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