Let \(A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]\) and \(P…
Let \(A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]\) and \(P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{array}\right]\). The sum of the prime factors of \(\left|P^{-1} A P-2 I\right|\) is equal to
[JEE Main 2024, 29 Jan (Shift 2)]
\(26\)
\(\left|\mathrm{P}^{-1} \mathrm{AP}-2 \mathrm{I}\right|=\left|\mathrm{P}^{-1} \mathrm{AP}-2 \mathrm{P}^{-1} \mathrm{P}\right|\)
\(=\left|\mathrm{P}^{-1}(\mathrm{~A}-2 \mathrm{I}) \mathrm{P}\right|\)
\(=\left|\mathrm{P}^{-1}\right||\mathrm{A}-2 \mathrm{I}||\mathrm{P}|\)
\(=|\mathrm{A}-2 \mathrm{I}|\)
\(=\left|\begin{array}{ccc}0 & 1 & 2 \\ 6 & 0 & 11 \\ 3 & 3 & 0\end{array}\right|\)
\(=69\)
So, Prime factor of 69 is 3 and 23
So, sum \(=26\)
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