Mathematics - Matrices Question with Solution | TestHub

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

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

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

Multiplying with $P$ on both sides

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

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

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

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

$\Rightarrow \mathrm{Det} M={\left(\mathrm{Det}\left(A-I\right)\right)}^2$

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

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

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

$\Rightarrow \alpha =36$