Mathematics Exam > Mathematics Questions > The set of all non-singular square matrices o...
Start Learning for Free
The set of all non-singular square matrices of same order with respect to matrix multiplication is
- a)quasi-group
- b)monoid
- c)group
- d)abelian group
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
The set of all non-singular square matrices of same order with respect...
Explanation:
To determine whether the set of all non-singular square matrices of the same order form a group with respect to matrix multiplication, we need to check whether it satisfies the four group axioms:
1. Closure: The product of any two non-singular square matrices of the same order is also a non-singular square matrix of the same order. Therefore, the set is closed under matrix multiplication.
2. Associativity: Matrix multiplication is associative, which means that for any three matrices A, B, and C of the same order, (AB)C = A(BC). Since matrix multiplication is associative, the set satisfies the associativity property.
3. Identity element: The identity matrix I, which is a non-singular square matrix of the same order as any matrix in the set, serves as the identity element. For any matrix A in the set, AI = A and IA = A. Therefore, the set contains an identity element.
4. Inverse element: For every non-singular square matrix A in the set, there exists an inverse matrix A^(-1) such that AA^(-1) = A^(-1)A = I, where I is the identity matrix. The inverse of A is also a non-singular square matrix of the same order. Therefore, the set contains inverse elements for every matrix.
Since the set of all non-singular square matrices of the same order satisfies all four group axioms, it can be concluded that it forms a group with respect to matrix multiplication.
Hence, the correct answer is option 'C' - group.
To determine whether the set of all non-singular square matrices of the same order form a group with respect to matrix multiplication, we need to check whether it satisfies the four group axioms:
1. Closure: The product of any two non-singular square matrices of the same order is also a non-singular square matrix of the same order. Therefore, the set is closed under matrix multiplication.
2. Associativity: Matrix multiplication is associative, which means that for any three matrices A, B, and C of the same order, (AB)C = A(BC). Since matrix multiplication is associative, the set satisfies the associativity property.
3. Identity element: The identity matrix I, which is a non-singular square matrix of the same order as any matrix in the set, serves as the identity element. For any matrix A in the set, AI = A and IA = A. Therefore, the set contains an identity element.
4. Inverse element: For every non-singular square matrix A in the set, there exists an inverse matrix A^(-1) such that AA^(-1) = A^(-1)A = I, where I is the identity matrix. The inverse of A is also a non-singular square matrix of the same order. Therefore, the set contains inverse elements for every matrix.
Since the set of all non-singular square matrices of the same order satisfies all four group axioms, it can be concluded that it forms a group with respect to matrix multiplication.
Hence, the correct answer is option 'C' - group.
Free Test
FREE
| Start Free Test |
Community Answer
The set of all non-singular square matrices of same order with respect...
- Identify the Algebraic Structure:
- The set in question is all non-singular (invertible) square matrices of the same order under matrix multiplication.
- Non-singular matrices have multiplicative inverses by definition.
-
- Check Group Axioms:
- Closure: The product of two invertible matrices is invertible.
- Associativity: Matrix multiplication is associative.
- Identity Element: The identity matrix II serves as the multiplicative identity.
- Inverses: Every non-singular matrix has an inverse.
Since all group axioms are satisfied, the set forms a group. -
- Eliminate Incorrect Options:
- A (Quasi-group): A group is a quasi-group, but this is not the most specific answer.
- B (Monoid): A monoid lacks inverses for all elements; here, inverses exist.
- D (Abelian Group): Matrix multiplication is not commutative in general, so the group is non-abelian.
-
- Hence option C is correct
|
Explore Courses for Mathematics exam
|
|
Similar Mathematics Doubts
Question Description
The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer?.
The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer?.
Solutions for The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The set of all non-singular square matrices of same order with respect to matrix multiplication isa)quasi-groupb)monoidc)groupd)abelian groupCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics exam
|
|
Signup to solve all Doubts
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup