Let $$P = \begin{bmatrix}2 & \alpha & 3 \\-\alpha & 2 & 0\\3 & -2 & \alpha \end{bmatrix}$$,where $$\alpha$$ is a real number such that det(P) = cofactor of second diagonal element of P. Then det(adj ($$P^{-1}$$)) equals

det P = $$2\left(2\alpha\ \right)-\alpha\ \left(-\alpha\ ^2\right)+3\left(2\alpha\ -6\right)=\ \alpha\ ^3+10\alpha\ -18$$

cofactor of $$a_{22}=2\alpha\ -9$$

$$\alpha\ ^3+10\alpha\ -18=2\alpha\ -9$$

$$\alpha\ ^3+8\alpha\ -9=0$$

=> $$\alpha\ =1$$

det(P) = $$2\alpha\ -9\ =-7$$

$$\det\left(adj\left(P^{-1}\right)\right)=\left[\det\left(P^{-1}\right)\right]^{3-1}$$   since $$\det\left(adjA\right)=\left(\det A\right)^{n-1}$$

= $$\left[\frac{1}{\det\left(P\right)}\right]^2=\frac{1}{49}$$