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[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?
- a)both [S] and [D] are symmetric
- b)both [S] and [D] are skew-symmetric
- c)[S] is skew-symmetric and [D] is symmetric
- d)[S] is symmetric and [D] is skew-symmetric.
Correct answer is option 'D'. Can you explain this answer?
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Differences between [A] and [A]T
- [A] is the given square matrix which is neither symmetric nor skew-symmetric.
- [A]T is the transpose of [A], which means the rows of [A] become the columns of [A]T and vice versa.
Sum and Difference of Matrices
- [S] is defined as the product of [A] and [A]T, meaning each element of [S] is the dot product of the corresponding row of [A] and column of [A]T.
- [D] is defined as the difference between [A] and [A]T, meaning each element of [D] is the difference between the corresponding elements of [A] and [A]T.
Properties of [S] and [D]
- To determine the properties of [S] and [D], we need to analyze their symmetry.
- A square matrix is symmetric if it is equal to its transpose, i.e., [M] = [M]T.
- A square matrix is skew-symmetric if the transpose of the matrix is equal to the negation of the original matrix, i.e., [M] = -[M]T.
Analysis of [S]
- [S] = [A] [A]T, which means [S] is the product of [A] and its transpose.
- Since [A] is neither symmetric nor skew-symmetric, it cannot be equal to its transpose, which means [S] is not equal to [S]T.
- Therefore, [S] is not symmetric.
Analysis of [D]
- [D] = [A] - [A]T, which means [D] is the difference between [A] and its transpose.
- Since [A] is neither symmetric nor skew-symmetric, it cannot be equal to its transpose, which means [D] is not equal to -[D]T.
- Therefore, [D] is not skew-symmetric.
Conclusion
- Based on the analysis, both [S] and [D] do not possess the properties of symmetry or skew-symmetry.
- The correct answer is option D: [S] is symmetric and [D] is skew-symmetric.
- [A] is the given square matrix which is neither symmetric nor skew-symmetric.
- [A]T is the transpose of [A], which means the rows of [A] become the columns of [A]T and vice versa.
Sum and Difference of Matrices
- [S] is defined as the product of [A] and [A]T, meaning each element of [S] is the dot product of the corresponding row of [A] and column of [A]T.
- [D] is defined as the difference between [A] and [A]T, meaning each element of [D] is the difference between the corresponding elements of [A] and [A]T.
Properties of [S] and [D]
- To determine the properties of [S] and [D], we need to analyze their symmetry.
- A square matrix is symmetric if it is equal to its transpose, i.e., [M] = [M]T.
- A square matrix is skew-symmetric if the transpose of the matrix is equal to the negation of the original matrix, i.e., [M] = -[M]T.
Analysis of [S]
- [S] = [A] [A]T, which means [S] is the product of [A] and its transpose.
- Since [A] is neither symmetric nor skew-symmetric, it cannot be equal to its transpose, which means [S] is not equal to [S]T.
- Therefore, [S] is not symmetric.
Analysis of [D]
- [D] = [A] - [A]T, which means [D] is the difference between [A] and its transpose.
- Since [A] is neither symmetric nor skew-symmetric, it cannot be equal to its transpose, which means [D] is not equal to -[D]T.
- Therefore, [D] is not skew-symmetric.
Conclusion
- Based on the analysis, both [S] and [D] do not possess the properties of symmetry or skew-symmetry.
- The correct answer is option D: [S] is symmetric and [D] is skew-symmetric.
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[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?a)both [S] and [D] are symmetricb)both [S] and [D] are skew-symmetricc)[S] is skew-symmetric and [D] is symmetricd)[S] is symmetric and [D] is skew-symmetric.Correct answer is option 'D'. Can you explain this answer?
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[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?a)both [S] and [D] are symmetricb)both [S] and [D] are skew-symmetricc)[S] is skew-symmetric and [D] is symmetricd)[S] is symmetric and [D] is skew-symmetric.Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about [A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?a)both [S] and [D] are symmetricb)both [S] and [D] are skew-symmetricc)[S] is skew-symmetric and [D] is symmetricd)[S] is symmetric and [D] is skew-symmetric.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for [A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?a)both [S] and [D] are symmetricb)both [S] and [D] are skew-symmetricc)[S] is skew-symmetric and [D] is symmetricd)[S] is symmetric and [D] is skew-symmetric.Correct answer is option 'D'. Can you explain this answer?.
[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?a)both [S] and [D] are symmetricb)both [S] and [D] are skew-symmetricc)[S] is skew-symmetric and [D] is symmetricd)[S] is symmetric and [D] is skew-symmetric.Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about [A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?a)both [S] and [D] are symmetricb)both [S] and [D] are skew-symmetricc)[S] is skew-symmetric and [D] is symmetricd)[S] is symmetric and [D] is skew-symmetric.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for [A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?a)both [S] and [D] are symmetricb)both [S] and [D] are skew-symmetricc)[S] is skew-symmetric and [D] is symmetricd)[S] is symmetric and [D] is skew-symmetric.Correct answer is option 'D'. Can you explain this answer?.
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