5. An $n \times n$ matrix A is called skew-symmetric if $A^T = -A$. (a) Let B be an $n \times n$ matrix. Prove that $B - B^T$ is skew-symmetric. (b) Let B be an $n \times n$ matrix. Prove that $B + B^T$ is symmetric. (c) Let B be an $n \times n$ matrix. Prove that B can be written as the sum of a symmetric matrix and a skew-symmetric matrix. Also show that such decomposition is unique.
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To prove (c), we need to show that any matrix B can be written as the sum of a symmetric matrix and a skew-symmetric matrix. Show more…
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Show that $A=B+C .$ Together with the previous exercise, this shows that every $n \times n$ matrix $A$ can be written as the sum of a symmetric matrix and a skew-symmetric matrix.
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