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3 Define the set S of matrices by S = {A = (aij) € M₂ (R): a11 = a22, a12 = -a21}. It turns out that S is a ring, with the operations of matrix addition and multiplication. (a) Write down two examples of elements of S, and compute their sum and product. (b) Prove the additive and multiplicative closure laws for S.

3 Define the set S of matrices by S = {A = (aij) € M₂ (R): a11 = a22, a12 = -a21}. It turns out that S is a ring, with the operations of matrix addition and multiplication. (a) Write down two examples of elements of S, and compute their sum and product. (b) Prove the additive and multiplicative closure laws for S.

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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Part A and B, written as clearly as possible please.
3 Define the set S of matrices by S = {A = (aij) = M₂ (R): a11 = a22, a12 = -a21}. It turns out that S is a ring, with the operations of matrix addition and multiplication. (a) Write down two examples of elements of S, and compute their sum and product. (b) Prove the additive and multiplicative closure laws for S. (c) Assume that M₂ (R) has already been proved to be a ring. If you wanted to prove that S was a ring, which axioms would follow immediately from the fact that M₂ (R) is a ring, and for which axioms would there be something else to check?
Transcribed Image Text:3 Define the set S of matrices by S = {A = (aij) = M₂ (R): a11 = a22, a12 = -a21}. It turns out that S is a ring, with the operations of matrix addition and multiplication. (a) Write down two examples of elements of S, and compute their sum and product. (b) Prove the additive and multiplicative closure laws for S. (c) Assume that M₂ (R) has already been proved to be a ring. If you wanted to prove that S was a ring, which axioms would follow immediately from the fact that M₂ (R) is a ring, and for which axioms would there be something else to check?
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