Question 5.and det { A} E Q (the rational numbers).Let (G, ) denote the set of all 2 x 2 real matrices A with det{A} 0(a) Prove that (G, ·) is a group with respect to multiplication. (Matrix multiplication is alwaysassociative, so you may assume that. But check closure and the existence of an identity elementand inverse elements very carefully.)(b) Is this group Abelian? Justify.

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Answered: Let (G, ) denote the set of all 2 x 2…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 16E: For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have...

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Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.
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