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The inverse of a symmetric matrix is
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Symmetric
Let $A$ be a symmetric matrix.
Then $A A^{-1}=I \Rightarrow\left(A A^{-1}\right)^T=I$
$\Rightarrow\left(A^{-1}\right)^T A^T=I \Rightarrow\left(A^{-1}\right)^T=\left(A^T\right)^{-1}$
$\Rightarrow\left(A^{-1}\right)^T=(A)^{-1},\left(\because A^T=A\right)$
$\Rightarrow A^{-1}$ is a symmetric matrix.
Then $A A^{-1}=I \Rightarrow\left(A A^{-1}\right)^T=I$
$\Rightarrow\left(A^{-1}\right)^T A^T=I \Rightarrow\left(A^{-1}\right)^T=\left(A^T\right)^{-1}$
$\Rightarrow\left(A^{-1}\right)^T=(A)^{-1},\left(\because A^T=A\right)$
$\Rightarrow A^{-1}$ is a symmetric matrix.
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