Matrices and Determinants: JEE Main 2021 Mathematics question

Let $P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {60} & {14} \cr } } \right]$ and $A = \left[ {\matrix{ 2 & 7 & {{\omega ^2}} \cr { - 1} & { - \omega } & 1 \cr 0 & { - \omega } & { - \omega + 1} \cr } } \right]$ where $\omega = {{ - 1 + i\sqrt 3 } \over 2}$, and I₃ be the identity matrix of order 3. If the determinant of the matrix (P^$-$1AP$-$I₃)² is $\alpha$$\omega$², then the value of $\alpha$ is equal to ______________.