\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Show that for every \(A \in M_n(\mathbb{R})\), the matrix \(\frac{1}{2}(A + A^\top)\) is symmetric and the matrix \(\frac{1}{2}(A - A^\top)\) is antisymmetric. Deduce that every square matrix can be written as a sum of a symmetric and an antisymmetric matrix.
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