Matrix group

Question

Can anybody share the info about the group/semi group of the matrices? and this problem..?

Consider the set H of all 3 × 3 matrices of the type:

$\begin{bmatrix} a & f &e \\ 0&b &d \\ 0& 0& c \end{bmatrix}$

where a, b, c, d, e and f are real numbers and abc ≠ 0. Under the matrix multiplication operation, the set H is:

(a) a group

(b) a monoid but not a group

(c) a semigroup but not a monoid

(d) neither a group nor a semigroup

Comments

There is no "group/semi-group" for matrices. Just apply the semi-group, group properties to a set whose elements are matrices of the form given. You should get the answer..

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I could not understand why it is not monoid. Because there exist an identity Matrix for the given matrix.

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yeah.. i just copy pasted the expected answer. it may be wrong.

Answer 1

Lets understand the definitions first,

Semi-Group : Satisfy {Closure, Associativity}

Monoid : Satisfy {Closure, Associativity, Identity}

Group : Satisfy {Closure, Associativity, Identity, Inverse}

The Given matrix is an example of upper triangular matrix. Which satisfy both closure as well as associativity property. Because multiplication of two upper triangular matrix is upper triangular matrix.

It has Identity Matrix I.  It also has Inverse because Inverse of an upper triangular matrix is also upper triangular matrix (Read this article) and given that none of the diagonal elements are zero (See here) . 

Hence answer should be A) a group 

Comments

Thank you for the explanation.

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That's how to answer this question. But do inverse exist for all upper triangular matrices?

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@rude Loyal i know that diagonal matrices are identity matrices. But how this triangular matrix is identity matrix? Will you please explain?

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@arjun Sir Thank you so very much for the appreciation. Its given that $abc != 0$, Hence There will not be any matrix which have detrminats 0. Hence Inverse for all the matrix will be available.

@padmaja your welcome.

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Its depends on cases.. Some triangular matrices does not inverse. But in some cases it do exist.

ex:  inverse of

1 0 0 

2 1 0

3 2 1    is

1 0 0

-2 1 0

-3 -2 1

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yes, but for given matrix - inverse always exist..