For a natural number n, letC, = {(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 2A1 = |1 2 020 1(0 1 2)A2 = |2 0 11 2 0(b) Let A be the matrix you chose in part (a). Compute A?. Check that A? e C3, andwrite out a brief explanation (one or two sentences is enough) explaining howyou 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.

Given that, Cn={aij∈MnR:ai,j=akl where, j-i ≡nl-k

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.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Matrices

Section2.1: Operations With Matrices

Problem 77E

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