Answered: 1) a) Let's start with 2x2 matrices.…

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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a) Let's start with 2x2 matrices. We have addition and scalar multiplication defined for these objects. Explain how to do both for these objects to someone who is new to matrices.

b) Do we have closure under these operations? Explain what closure under the operations means and then explain why/why not each operation is/is not closed.

c) Does it seem reasonable that the 8 axioms will hold for these objects? If not find one that you think will not work and show it does not work (you an use an actual example with numbers). If so pick 2 of the axioms and explain how they hold for the objects (do not use numbers in your matrices- these should hold for ALL 2x2 matrices remember). 8 axioms are attached below.

1. Associativity: u + (v + w) = (u + v) + w
2. Commutativity: u + v = v + u
3. Zero element: There is an element 0 e V, such that 0 + v = v + 0 = v.
4. Inverse: For every element v in v, there is an element –v in V such that v + (-v) = 0
5. Compatibility of scalar multiplication: For every scalars a, b and a vector v,
a (bv) = (ab)v.
6. Identity element of scalar multiplication: 1v = v, where 1 is the multiplicative identity
of the field.
7. Distributive property: a (u + v) = au + av
8. (a + b)v = av + bv

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