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Q.
The inverse of a symmetric matrix (if it
exists) is
Matrices
Solution:
Let $A$ be an invertible symmetric matrix.
$\text { We have } A A^{-1}=A^{-1} A=I_n $
$\Rightarrow \left(A A^{-1}\right)^{\prime}=\left(A^{-1}A\right)^{\prime}=\left(I_n\right)^{\prime} $
$\Rightarrow \left(A^{-1}\right)^{\prime}A^{\prime}=A^{\prime}\left(A^{-1}\right)^{\prime}=I_n$
$\Rightarrow \left(A^{-1}\right)^{\prime} A=A\left(A^{-1}\right)^{\prime}=I_n$
$\left(A^{-1}\right)^{\prime}=A^{-1}$ [inverse of a matrix is unique]