Question:
If $A$ is a non-singular symmetric matrix, write whether $A^{-1}$ is symmetric or skew-symmetric.
Solution:
Let $A$ be an invertible symmetric matrix. Then,
$|A| \neq 0$ and $A^{T}=A$
Now, $\left(A^{-1}\right)^{T}=\left(A^{T}\right)^{-1}$
$\Rightarrow\left(A^{-1}\right)^{T}=A^{-1} \quad\left[\because A^{T}=A\right]$
Thus, $A^{-1}$ is symmetric matrix.
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