Given:

$A=\left[\begin{array}{ccc}7& 3& \alpha \\ \beta & 1& -11\\ -5& \gamma & 19\end{array}\right]$

And,

$A\left[\begin{array}{c}5\\ -13\\ 11\end{array}\right]=\left[\begin{array}{c}-290\\ -119\\ 210\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}7& 3& \alpha \\ \beta & 1& -11\\ -5& \gamma & 19\end{array}\right]\left[\begin{array}{c}5\\ -13\\ 11\end{array}\right]=\left[\begin{array}{c}-290\\ -119\\ 210\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}11\alpha -4\\ 5\beta -134\\ 184-13\gamma \end{array}\right]=\left[\begin{array}{c}-290\\ -119\\ 210\end{array}\right]$

So,

$11\alpha -4=-290\Rightarrow \alpha =-26$

$5\beta -134=-19\Rightarrow \beta =3$

$184-13\gamma =210\Rightarrow \gamma =-2$

So,

$A=\left[\begin{array}{ccc}7& 3& -26\\ 3& 1& -11\\ -5& -2& 19\end{array}\right]$

$\Rightarrow \left|A\right|=7\left(-3\right)-3\left(2\right)-26\left(-1\right)=1$

$\mathrm{adj}\left(A\right)={\left[\begin{array}{ccc}-3& -2& -1\\ -5& 3& -1\\ -7& -1& -2\end{array}\right]}^{\mathrm{T}}$

$\Rightarrow \mathrm{adj}\left(A\right)=\left[\begin{array}{ccc}-3& -5& -7\\ -2& 3& -1\\ -1& -1& -2\end{array}\right]$

So,

$A^{-1}=\frac{\mathrm{adj}\left(A\right)}{\left|A\right|}=\left[\begin{array}{ccc}-3& -5& -7\\ -2& 3& -1\\ -1& -1& -2\end{array}\right]$

So,

$\mathrm{adj}\left(A^{-1}\right)={\left[\begin{array}{ccc}-7& -3& 5\\ -3& -1& 2\\ 26& 11& -19\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{ccc}-7& -3& 26\\ -3& -1& 11\\ 5& 2& -19\end{array}\right]$

Also,

$\left|\mathrm{adj}\left(A\right)\right|=1$

$\mathrm{adj}\left(\mathrm{adj}\left(A\right)\right)=\left[\begin{array}{ccc}-7& -3& 26\\ -3& -1& 11\\ 5& 2& -19\end{array}\right]$

So,

${\left(\mathrm{adj}\left(A\right)\right)}^{-1}=\frac{\mathrm{adj}\left(\mathrm{adj}\left(A\right)\right)}{\left|\mathrm{adj}\left(A\right)\right|}=\left[\begin{array}{ccc}-7& -3& 26\\ -3& -1& 11\\ 5& 2& -19\end{array}\right]$

Hence,

${\left(\mathrm{adj}\left(A\right)\right)}^{-1}+\left(\mathrm{adj}\left(A^{-1}\right)\right)=2\left[\begin{array}{ccc}-7& -3& 26\\ -3& -1& 11\\ 5& 2& -19\end{array}\right]$

$\Rightarrow {\left(\mathrm{adj}\left(A\right)\right)}^{-1}+\left(\mathrm{adj}\left(A^{-1}\right)\right)=-2\left[\begin{array}{ccc}7& 3& -26\\ 3& 1& -11\\ -5& -2& 19\end{array}\right]$

$\Rightarrow {\left(\mathrm{adj}\left(A\right)\right)}^{-1}+\left(\mathrm{adj}\left(A^{-1}\right)\right)=-2A$