Let P = [ cases - 30 & 20 & 56 90 & 140 & 112 120 & 60 & 14 cases ]...

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

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

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

Multiplying with $P$ on both sides

$⇒ PM = {A}^{2}P - 2AP + P$

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

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

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

$⇒ DetM = {\left(Det\left(A - I\right)\right)}^{2}$

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

Det \left(A - I\right) = \left({w}^{2} + w + w\right) + 7\left(- w\right) + {w}^{3} = - 6w$</p><p>Det {\left(\left(A - I\right)\right)}^{2} = 36{w}^{2}$

$⇒ \alpha = 36$