Mathematics Exam > Mathematics Questions > Which of the following(s) is/are correct?a)Th...
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Which of the following(s) is/are correct?
- a)The transpose of a symmetric matrix need not be summetric matrix.
- b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.
- c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.
- d)If A is any square matrix, then A + A'is always symmetric
Correct answer is option 'B,C,D'. Can you explain this answer?
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Which of the following(s) is/are correct?a)The transpose of a symmetri...
Statement a: The transpose of a symmetric matrix need not be symmetric matrix.
Statement b: If A and B are symmetric matrix of same order, then AB BA must be symmetric matrix.
Statement c: If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.
Statement d: If A is any square matrix, then A A is always symmetric.
Explanation:
Statement a: The transpose of a symmetric matrix need not be symmetric matrix.
A matrix is said to be symmetric if it is equal to its transpose. So, if a matrix A is symmetric, then A^T = A. However, the transpose of a symmetric matrix need not be symmetric. For example, consider the matrix A = [1 2; 2 3]. A^T = [1 2; 2 3] which is not equal to A. Therefore, statement a is correct.
Statement b: If A and B are symmetric matrix of same order, then AB BA must be symmetric matrix.
If A and B are symmetric matrices of the same order, then AB and BA may or may not be symmetric matrices. For example, consider the matrices A = [1 2; 2 3] and B = [3 4; 4 5]. AB = [11 14; 18 23], which is not equal to its transpose, and BA = [7 10; 10 14], which is also not equal to its transpose. Therefore, statement b is incorrect.
Statement c: If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.
If A is a symmetric matrix, then all positive integral powers of A will also be symmetric matrices. This can be proved by induction.
For the base case, A^1 = A, which is symmetric by definition.
Assuming that A^n is symmetric for some positive integer n, we need to show that A^(n+1) is also symmetric.
A^(n+1) = A^n * A = (A^n)^T * A = (A^T)^n * A
Since A is symmetric, A^T = A, so we can write (A^T)^n * A as A^n * A = A^(n+1).
Therefore, A^(n+1) is symmetric.
By induction, all positive integral powers of A are symmetric matrices. Therefore, statement c is correct.
Statement d: If A is any square matrix, then A A is always symmetric.
If A is any square matrix, then A * A is always symmetric. This can be easily verified by taking the transpose of A * A.
(A * A)^T = A^T * A^T = A * A
Therefore, A * A is symmetric. Therefore, statement d is correct.
To summarize:
- Statement a is correct.
- Statement b is incorrect.
- Statement c is correct.
- Statement d is correct.
Statement b: If A and B are symmetric matrix of same order, then AB BA must be symmetric matrix.
Statement c: If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.
Statement d: If A is any square matrix, then A A is always symmetric.
Explanation:
Statement a: The transpose of a symmetric matrix need not be symmetric matrix.
A matrix is said to be symmetric if it is equal to its transpose. So, if a matrix A is symmetric, then A^T = A. However, the transpose of a symmetric matrix need not be symmetric. For example, consider the matrix A = [1 2; 2 3]. A^T = [1 2; 2 3] which is not equal to A. Therefore, statement a is correct.
Statement b: If A and B are symmetric matrix of same order, then AB BA must be symmetric matrix.
If A and B are symmetric matrices of the same order, then AB and BA may or may not be symmetric matrices. For example, consider the matrices A = [1 2; 2 3] and B = [3 4; 4 5]. AB = [11 14; 18 23], which is not equal to its transpose, and BA = [7 10; 10 14], which is also not equal to its transpose. Therefore, statement b is incorrect.
Statement c: If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.
If A is a symmetric matrix, then all positive integral powers of A will also be symmetric matrices. This can be proved by induction.
For the base case, A^1 = A, which is symmetric by definition.
Assuming that A^n is symmetric for some positive integer n, we need to show that A^(n+1) is also symmetric.
A^(n+1) = A^n * A = (A^n)^T * A = (A^T)^n * A
Since A is symmetric, A^T = A, so we can write (A^T)^n * A as A^n * A = A^(n+1).
Therefore, A^(n+1) is symmetric.
By induction, all positive integral powers of A are symmetric matrices. Therefore, statement c is correct.
Statement d: If A is any square matrix, then A A is always symmetric.
If A is any square matrix, then A * A is always symmetric. This can be easily verified by taking the transpose of A * A.
(A * A)^T = A^T * A^T = A * A
Therefore, A * A is symmetric. Therefore, statement d is correct.
To summarize:
- Statement a is correct.
- Statement b is incorrect.
- Statement c is correct.
- Statement d is correct.
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Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + Ais always symmetricCorrect answer is option 'B,C,D'. Can you explain this answer?
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Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + Ais always symmetricCorrect answer is option 'B,C,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 Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + Ais always symmetricCorrect answer is option 'B,C,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 Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + Ais always symmetricCorrect answer is option 'B,C,D'. Can you explain this answer?.
Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + Ais always symmetricCorrect answer is option 'B,C,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 Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + Ais always symmetricCorrect answer is option 'B,C,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 Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + Ais always symmetricCorrect answer is option 'B,C,D'. Can you explain this answer?.
Solutions for Which of the following(s) is/are correct?a)The transpose of a symmetric matrix need not be summetric matrix.b)If A and B are symmetric matrix of same order, then AB + BA must be symmetric matrix.c)If A is symmetric matrix, then all positive integral powers of A are symmetric matrices.d)If A is any square matrix, then A + Ais always symmetricCorrect answer is option 'B,C,D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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