The inverse of a symmetric matrix is

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(A) Let $A$ be a symmetric matrix.By definition of a symmetric matrix,$A^T = A$.We know that $A A^{-1} = I$.Taking the transpose on both sides,we get

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(A) Let $A$ be a symmetric matrix.By definition of a symmetric matrix,$A^T = A$.We know that $A A^{-1} = I$.Taking the transpose on both sides,we get $(A A^{-1})^T = I^T$.Using the property $(AB)^T = B^T A^T$,we have $(A^{-1})^T A^T = I$.Since $A^T = A$,this becomes $(A^{-1})^T A = I$.Multiplying both sides by $A^{-1}$ on the right,we get $(A^{-1})^T A A^{-1} = I A^{-1}$.$(A^{-1})^T I = A^{-1}$.$(A^{-1})^T = A^{-1}$.Since the transpose of $A^{-1}$ is equal to $A^{-1}$,the inverse of a symmetric matrix is symmetric.

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