$=\left( P ^{-1} AP \right)^{2}-2 P ^{-1} AP + I$

$= P ^{-1} A ^{2} P -2 P^{-1} AP + I$

$PM = A ^{2} P -2 AP + P$

$=\left( A ^{2}-2 A . I + I ^{2}\right) P$

$\Rightarrow \quad \operatorname{Det}( PM )=\operatorname{Det}\left(( A - I )^{2} \times P \right)$

$\Rightarrow \quad \operatorname{DetP.DetM}=\operatorname{Det}( A - I )^{2} \times \operatorname{Det}( P )$

$\Rightarrow \quad \operatorname{Det} M =(\operatorname{Det}( A - I ))^{2}$

Now $A-I=\left[\begin{array}{ccc}1 & 7 & w^{2} \\ -1 & -w-1 & 1 \\ 0 & -w & -w\end{array}\right]$

$\operatorname{Det}( A - I )=\left( w ^{2}+ w + w \right)+7(- w )+ w ^{3}=-6 w$

$\operatorname{Det}(( A - I ))^{2}=36 w ^{2}$

$\Rightarrow \alpha=36$