JEE 2024 — Mathematics PYQ

$LetA=\left[\begin{array}{ccc}2& amp;1& amp;2\\ 6& amp;2& amp;11\\ 3& amp;3& amp;2\end{array}\right]\text{ and }P=\left[\begin{array}{ccc}1& amp;2& amp;0\\ 5& amp;0& amp;2\\ 7& amp;1& amp;5\end{array}\right]$

$letB=P^{-1}AP-2I$
$PostmultiplybyP\Rightarrow PB=AP-2P\{\because P{P}^{-1}=I\&PI=P\}$
$Postmultiplyby{P}^{-1}\Rightarrow PB{P}^{-1}=A-2I$
$Now|PB{P}^{-1}|=|A-2I|$
$|P|.|B|.|P^{-1}|=\left|\begin{array}{ccc}2-2& amp;1& amp;2\\ 6& amp;2-2& amp;11\\ 3& amp;3& amp;2-2\end{array}\right|$

$PB=AP-2P\{\because P{P}^{-1}=I\&PI=P\}$
$PB{P}^{-1}=A-2I\Rightarrow |PB{P}^{-1}|=|A-2I|$
$|P|.|B|.\frac{1}{|P|}=\left|\begin{array}{ccc}0& amp;1& amp;2\\ 6& amp;0& amp;11\\ 3& amp;3& amp;0\end{array}\right|$
$|B|=-1(-33)+2(18)\Rightarrow |P^{-1}AP-2I|=33+36=69$
$Primefactorsof69is3\&23\Rightarrow sum=3+23=26$