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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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Prove or disprove that the set G in Exercise 38 is a group with respect to addition.

38. Let G be the set of all matrices in M 3 ( ) that have the form

[ a 0 0 0 b 0 0 0 c ] with all three numbers a , b , and c nonzero. Prove or disprove that G is a group with respect to multiplication.

To determine

Whether the set G of all matrices in M3() that have the form (a000b000c) with all three numbers a,b, and c nonzero is a group with respect to addition.

Explanation

Given information:

Let G be a set of all matrices in M3() that have the form (a000b000c) with all three numbers a,b, and c nonzero and operation addition.

Explanation:

Definition of a group:

Suppose the binary operation is defined for element of set G. Then G is a group with respect to , provided the following conditions hold:

1. G is closed under . That is, xG and yG imply that xy is in G.

2. is associative. For all x,y,z in G, x(yz)=(xy)z.

3. G has an identity element e. There is an e in G such that xe=ex=x for all xG.

4. G contains inverses. For each aG, there exists bG such that ab=ba=e.

Consider the set G={(a000b000c)M3()|a,b,c0} with operation addition.

First condition:

Let (200010001),(200010002)G

(200010001)+(200010002)=(2+(2)0001+10001+2)=(000000003) G,

Hence, G is not closed under addition

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