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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Let A and B be two symmetric matrices of order 3.
Assertion (A): A(BA) and (AB)A are symmetric matrices.
Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.
Let A and B be two symmetric matrices of order 3.
Assertion (A): A(BA) and (AB)A are symmetric matrices.
Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.
- a)Both A and R are true and R is the correct explanation of A
- b)Both A and R are true but R is NOT the correct explanation of A
- c)A is true but R is false
- d)A is false and R is True
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Directions : In the following questions, A statement of Assertion (A) ...
Generally (AB)’ = B’ A’
If AB = BA, then (AB)’ = (BA)’ = A’ B’ = AB
Since (AB)’ = AB, AB is a symmetric matrix. Hence R is true.
A(BA) = (AB)A = ABA
(ABA)’ = A’ B’ A’ = ABA.
A(BA) and (AB)A are symmetric matrices. Hence A is true.
But R is not the correct explanation for A.
If AB = BA, then (AB)’ = (BA)’ = A’ B’ = AB
Since (AB)’ = AB, AB is a symmetric matrix. Hence R is true.
A(BA) = (AB)A = ABA
(ABA)’ = A’ B’ A’ = ABA.
A(BA) and (AB)A are symmetric matrices. Hence A is true.
But R is not the correct explanation for A.
Most Upvoted Answer
Directions : In the following questions, A statement of Assertion (A) ...
Assertion (A): A(BA) and (AB)A are symmetric matrices.
Reason (R): AB is a symmetric matrix if the matrix multiplication of A with B is commutative.
Explanation:
To prove this assertion, we need to show that both statements A and R are true and that statement R is the correct explanation of statement A.
Statement A: A(BA) and (AB)A are symmetric matrices.
Let's consider the matrix A and B to be symmetric matrices of order 3.
A matrix is said to be symmetric if it is equal to its transpose. In other words, for a matrix A, A = A^T.
Now, let's analyze the matrix products A(BA) and (AB)A.
Matrix Product A(BA):
A(BA) = A(BA)^T [Since A(BA) is symmetric, it is equal to its transpose]
= A(AB)^T [Since (BA)^T = (AB)^T]
= (A^T)(AB) [Using the property of transpose]
= (AB)A [Since A^T = A]
Therefore, A(BA) is a symmetric matrix.
Matrix Product (AB)A:
(AB)A = (AB)A^T [Since (AB)A is symmetric, it is equal to its transpose]
= AB^TA^T [Using the property of transpose]
= AB^TA [Since A^T = A]
Therefore, (AB)A is a symmetric matrix.
Statement R: AB is a symmetric matrix if the matrix multiplication of A with B is commutative.
A matrix product AB is commutative if AB = BA, i.e., the order of multiplication does not matter.
If AB is commutative, then (AB)A = A(BA) = A^T(BA) = (BA)A = BA^TA = BA.
Thus, if AB is commutative, then both A(BA) and (AB)A are symmetric matrices.
Conclusion:
From the above analysis, we can conclude that both statement A and statement R are true. Furthermore, statement R is the correct explanation of statement A, as the commutativity of matrix multiplication (AB = BA) ensures that both A(BA) and (AB)A are symmetric matrices. Hence, the correct answer is option 'B'.
Reason (R): AB is a symmetric matrix if the matrix multiplication of A with B is commutative.
Explanation:
To prove this assertion, we need to show that both statements A and R are true and that statement R is the correct explanation of statement A.
Statement A: A(BA) and (AB)A are symmetric matrices.
Let's consider the matrix A and B to be symmetric matrices of order 3.
A matrix is said to be symmetric if it is equal to its transpose. In other words, for a matrix A, A = A^T.
Now, let's analyze the matrix products A(BA) and (AB)A.
Matrix Product A(BA):
A(BA) = A(BA)^T [Since A(BA) is symmetric, it is equal to its transpose]
= A(AB)^T [Since (BA)^T = (AB)^T]
= (A^T)(AB) [Using the property of transpose]
= (AB)A [Since A^T = A]
Therefore, A(BA) is a symmetric matrix.
Matrix Product (AB)A:
(AB)A = (AB)A^T [Since (AB)A is symmetric, it is equal to its transpose]
= AB^TA^T [Using the property of transpose]
= AB^TA [Since A^T = A]
Therefore, (AB)A is a symmetric matrix.
Statement R: AB is a symmetric matrix if the matrix multiplication of A with B is commutative.
A matrix product AB is commutative if AB = BA, i.e., the order of multiplication does not matter.
If AB is commutative, then (AB)A = A(BA) = A^T(BA) = (BA)A = BA^TA = BA.
Thus, if AB is commutative, then both A(BA) and (AB)A are symmetric matrices.
Conclusion:
From the above analysis, we can conclude that both statement A and statement R are true. Furthermore, statement R is the correct explanation of statement A, as the commutativity of matrix multiplication (AB = BA) ensures that both A(BA) and (AB)A are symmetric matrices. Hence, the correct answer is option 'B'.
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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?
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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?.
Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?.
Solutions for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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ample number of questions to practice Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Let A and B be two symmetric matrices of order 3.Assertion (A): A(BA) and (AB)A are symmetric matrices.Reason (R): AB is symmetric matrix if matrix multiplication of A with B is commutative.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice JEE tests.
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