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Consider the following statements in respect of symmetric matrices A and B:
1. AB is symmetric.
2. A2 + B2 is symmetric.
2. A2 + B2 is symmetric.
Which of the above statements is/are correct ?
- a)1 only
- b)2 only
- c)Both 1 and 2
- d)Neither 1 nor 2
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Consider the following statements in respect of symmetric matrices A a...
Statement Analysis:
We are given two statements regarding symmetric matrices A and B. We need to determine which of the statements are correct.
Statement 1:
AB is symmetric.
Statement 2:
A^2 * B^2 is symmetric.
Explanation:
To determine the correctness of the statements, let's consider the properties of symmetric matrices.
Properties of Symmetric Matrices:
1. A matrix A is symmetric if and only if A = A^T, where A^T denotes the transpose of matrix A.
2. If A and B are both symmetric matrices of the same order, then AB is symmetric if and only if AB = BA.
Now, let's analyze each statement:
Statement 1: AB is symmetric.
To prove this statement, we need to show that (AB)^T = AB.
Let's assume matrix A is of order n x n and matrix B is of order n x m.
(AB)^T = B^T * A^T [Transpose of a product of matrices]
= B * A [Since both matrices A and B are symmetric]
Since AB = BA (by property 2), we can conclude that AB is symmetric. Hence, statement 1 is correct.
Statement 2: A^2 * B^2 is symmetric.
To prove this statement, we need to show that (A^2 * B^2)^T = A^2 * B^2.
(A^2 * B^2)^T = (B^2)^T * (A^2)^T [Transpose of a product of matrices]
= B^2 * A^2 [Since both matrices A and B are symmetric]
Since A^2 * B^2 ≠ B^2 * A^2 (unless A and B commute), we cannot conclude that A^2 * B^2 is symmetric. Hence, statement 2 is incorrect.
Conclusion:
Based on the analysis, we can conclude that statement 1 is correct, but statement 2 is incorrect. Therefore, the correct answer is option 'B' (2 only).
We are given two statements regarding symmetric matrices A and B. We need to determine which of the statements are correct.
Statement 1:
AB is symmetric.
Statement 2:
A^2 * B^2 is symmetric.
Explanation:
To determine the correctness of the statements, let's consider the properties of symmetric matrices.
Properties of Symmetric Matrices:
1. A matrix A is symmetric if and only if A = A^T, where A^T denotes the transpose of matrix A.
2. If A and B are both symmetric matrices of the same order, then AB is symmetric if and only if AB = BA.
Now, let's analyze each statement:
Statement 1: AB is symmetric.
To prove this statement, we need to show that (AB)^T = AB.
Let's assume matrix A is of order n x n and matrix B is of order n x m.
(AB)^T = B^T * A^T [Transpose of a product of matrices]
= B * A [Since both matrices A and B are symmetric]
Since AB = BA (by property 2), we can conclude that AB is symmetric. Hence, statement 1 is correct.
Statement 2: A^2 * B^2 is symmetric.
To prove this statement, we need to show that (A^2 * B^2)^T = A^2 * B^2.
(A^2 * B^2)^T = (B^2)^T * (A^2)^T [Transpose of a product of matrices]
= B^2 * A^2 [Since both matrices A and B are symmetric]
Since A^2 * B^2 ≠ B^2 * A^2 (unless A and B commute), we cannot conclude that A^2 * B^2 is symmetric. Hence, statement 2 is incorrect.
Conclusion:
Based on the analysis, we can conclude that statement 1 is correct, but statement 2 is incorrect. Therefore, the correct answer is option 'B' (2 only).
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Consider the following statements in respect of symmetric matrices A a...
Correct option is B. A2 + B2 is symmetric.
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Consider the following statements in respect of symmetric matrices A and B:1. AB is symmetric.2. A2 + B2 is symmetric.Which of the above statements is/are correct ?a)1 onlyb)2 onlyc)Both 1 and 2d)Neither 1 nor 2Correct answer is option 'B'. Can you explain this answer?
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