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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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Let G be the set of all matrices in M 3 ( ) that have the form

[ 1 a b 0 1 c 0 0 1 ] for arbitrary real numbers a , b , and c . 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 (1ab01c001) for arbitrary real numbers a,b, and c is a group with respect to multiplication.

Explanation

Given information:

The set G of all matrices in M3() that have the form (1ab01c001) for arbitrary real numbers a,b, and c.

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={(1ab01c001)M3()|a,b,c} with operation multiplication.

First condition:

Let (1ab01c001),(1de01f001)G such that a,b,c,d,e,f are arbitrary real numbers.

(1ab01c001).(1de01f001)=(1a+de+af+b01c+f001)G

Hence, G is closed under multiplication.

Second condition:

Let (1ab01c001),(1de01f001),(1gh01i001)G; then

(1ab01c001)((1de01f001)(1gh01i001))=((1ab01c001)(1de01f001))(1gh01i001)

As matrix multiplication is associative, multiplication is associ

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