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* The sum and difference of two symmetric matrices is symmetric.
* This is not always true for the [[matrix multiplication|product]]: given symmetric matrices <math>A</math> and <math>B</math>, then <math>AB</math> is symmetric if and only if <math>A</math> and <math>B</math> [[commutativity|commute]], i.e., if <math>AB=BA</math>.
* For any integer <math>n</math>, <math>A^n</math> is symmetric if <math>A</math> is symmetric.
* If <math>A^{-1}</math> exists, it is symmetric if and only if <math>A</math> is symmetric.
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=== Matrix congruent to a symmetric matrix ===
Any matrix [[matrix congruence|congruent]] to a symmetric matrix is again symmetric: if <math>X</math> is a symmetric matrix, then so is <math>A X A^{\mathrm T}</math> for any matrix <math>A</math>.
=== Symmetry implies normality ===
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