The set M of square matrices ( of same order) with respect to matrix multiplication is

a)
quasi-group

b)
group

c)
monoid

d)
semi-group

Correct answer is option 'C'. Can you explain this answer?

Question

The set M of square matrices ( of same order) with respect to matrix multiplication isa)quasi-groupb)groupc)monoidd)semi-groupCorrect answer is option 'C'. Can you explain this answer?

Answer

Explanation:
To determine the set M of square matrices with respect to matrix multiplication, we need to analyze the properties of this set.

Definition of a Monoid:
A monoid is a set equipped with an associative binary operation and an identity element.

Associative Binary Operation:
Matrix multiplication is an associative binary operation, which means that for any three matrices A, B, and C of the same order, the following holds:
(A * B) * C = A * (B * C)

Identity Element:
The identity element for matrix multiplication is the identity matrix. The identity matrix I is a square matrix with ones on the main diagonal and zeros elsewhere, such that for any matrix A of the appropriate size, the following holds:
A * I = I * A = A

Analysis:
Considering the properties of matrix multiplication, we can conclude the following:

1. Associativity: Matrix multiplication is associative, satisfying the requirement of an associative binary operation.

2. Identity Element: The identity matrix serves as the identity element for matrix multiplication, satisfying the requirement of an identity element.

Therefore, the set M of square matrices with respect to matrix multiplication forms a monoid.

Conclusion:
The correct answer is option 'C' - monoid. The set M of square matrices with respect to matrix multiplication satisfies the properties of an associative binary operation and has an identity element, making it a monoid.

Answer

The Answer should be D, since the set M of square matrices is closed under matrix multiplication, the operation is associative, and for each matrix A belonging to M, an identity matrix I exists such that AI = IA. Thus M is a monoid. Inverse element does not exist for all singular matrices.

Answer

Solution:

The set M of square matrices (of same order) with respect to matrix multiplication is a monoid.

A monoid is a set equipped with an associative binary operation and an identity element. In the case of square matrices, the binary operation is matrix multiplication, which is associative. The identity element is the identity matrix, a square matrix with ones on the diagonal and zeros elsewhere.

It is not a group because not every matrix has an inverse (a matrix that, when multiplied by the original matrix, yields the identity matrix).

It is not a quasi-group because although every element has a left and right inverse, these inverses are not necessarily the same.

It is a semi-group because it is a set with an associative binary operation, but this term is less specific than monoid since it doesn't specify the existence of an identity element.

So, the most specific and correct answer is monoid.

Answer

C

Answer

Option c

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Which of the following is true?

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