Let P = [-30, 20, 56; 90, 140, 112; 120, 60, 14] and A = [2, 7, ω^2; -1, -ω, 1; 0, -ω, -ω+1] where ω=(-1+i √ 3)/2, and I_3 be the identity matrix of…

Let $\displaystyle \mathrm{P}=\left[\begin{array}{rrr}-30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14\end{array}\right]$ and $\displaystyle \mathrm{A}=\left[\begin{array}{rrr}2 & 7 & \omega^2 \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega+1\end{array}\right]$ where $\displaystyle \omega=\frac{-1+i \sqrt{3}}{2}$, and $\displaystyle \mathrm{I}_3$ be the identity matrix of order 3. If the determinant of the matrix $\displaystyle \left(\mathrm{P}^{-1} \mathrm{AP}-\mathrm{I}_3\right)^2$ is $\displaystyle \alpha \omega^2$, then the value of $\displaystyle \alpha$ is equal to \_\_\_\_.