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In \(M_n(\mathbb{R})\), let \(\mathcal{S}_n = \{M \in M_n(\mathbb{R}) : M^\top = M\}\) (symmetric matrices) and \(\mathcal{A}_n = \{M \in M_n(\mathbb{R}) : M^\top = -M\}\) (skew-symmetric matrices). Show that \(\mathcal{S}_n \oplus \mathcal{A}_n = M_n(\mathbb{R})\).
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