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Consider the set H of all 3 × 3 matrices of the type GATECS2005Q46 where a, b, c, d, e and f are real numbers and abc ≠ 0. Under the matrix multiplication operation, the set H is

 

a group

 

a monoid but not a group

C

a semigroup but not a monoid

D

neither a group nor a semigroup

2 Answers

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a group..(it is closed, associative, unique identity and unique inverse..)
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1=>closure satisfy because given domain is real number and co domain also will be real number...

2=>associative property satisfy..

3=>unique identity will be satisfy ,,,because M*I=M=I*M(in the case of matrix multiplication, identity matrix(I) will be unique identity)...

4=>unique inverse will be satisfy....because det(m)=abc (given that it is !=0) so matrix will be non-singular matrix therefore unique inverse will exist ...

so it is group .......option (A) is correct answer

some addition information===>if a matrix is singular matrix then it will be monoid but not group,,,,,if it is non singular matrix then it will be group....but abelian property is neither satisfy by singular matrix nor by non-singular matrix ...because matrix multiplication never hold commutative property

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