$A=\frac{1}{7}\left[\begin{array}{ccc}3& -2& 6\\ -6& -3& 2\\ -2& 6& 3\end{array}\right]$

$\left|A\right|=\left[\frac{3}{7}\left(\frac{-21}{49}\right)+\frac{2}{7}\left(\frac{-14}{49}\right)+\frac{6}{7}\left(\frac{-42}{7}\right)\right]$

$=\frac{-9-4-36}{49}=-1$

$A^T=\frac{1}{7}\left[\begin{array}{ccc}3& -6& -2\\ -2& -3& 6\\ 6& 2& 3\end{array}\right]$

$A^{-1}=\frac{1}{\left|A\right|}Adj\left(A\right)$

Now $Adj\left(A\right)=\frac{1}{7}{\left[\begin{array}{ccc}-21& 14& -42\\ 42& 21& -14\\ 14& -42& -21\end{array}\right]}^T$

$=\frac{1}{7}\left[\begin{array}{ccc}-21& 42& 14\\ 14& 21& -42\\ -42& -14& -21\end{array}\right]$

$=\left[\begin{array}{ccc}-3& 6& 2\\ 2& 3& -6\\ -6& -2& -3\end{array}\right]$

Hence, $A^{-1}=\frac{1}{\left(-1\right)7}\left[\begin{array}{ccc}-3& 6& 2\\ 2& 3& -6\\ -6& -2& -3\end{array}\right]$

$=\frac{1}{7}\left[\begin{array}{ccc}3& -6& -2\\ -2& -3& 6\\ 6& 2& 3\end{array}\right]$