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

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

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.

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Step 1: Define the set S given in details.

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Step 2: We find two matrices from S and find their sum and product.

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Step 3: We prove the additive and multiplicative closure laws for S.

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Follow-up Question

can you help with part c) ?

(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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