The sum of two symmetric matrices is also a symmetric matrix.
Question
The sum of two symmetric matrices is also a symmetric matrix.
Solution
Yes, the statement is correct. The sum of two symmetric matrices is also a symmetric matrix. Here's the step by step explanation:
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Let's consider two symmetric matrices A and B. A matrix is symmetric if it is equal to its transpose. So, A = A^T and B = B^T.
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The sum of A and B is another matrix C, such that C = A + B.
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To check if C is also symmetric, we need to check if C is equal to its transpose.
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The transpose of C (C^T) is equal to the transpose of (A + B), which is A^T + B^T.
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Since A and B are symmetric, A^T = A and B^T = B. So, A^T + B^T = A + B = C.
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Therefore, C = C^T, which means that the sum of two symmetric matrices (C) is also a symmetric matrix.
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