Let $\mathrm{A}=\left[\begin{array}{ccc}2& 1& 2\\ 6& 2& 11\\ 3& 3& 2\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{lll}1& 2& 0\\ 5& 0& 2\\ 7& 1& 5\end{array}\right]$. The sum of the prime factors of $\left|{\mathrm{P}}^{-1}\mathrm{AP}-2\mathrm{I}\right|$ is equal to

Let, $\mathrm{y}=\left|{\mathrm{P}}^{-1}\mathrm{AP}-2\mathrm{I}\right|$

$\Rightarrow \mathrm{y}=\left|{\mathrm{P}}^{-1}\mathrm{AP}-2{\mathrm{P}}^{-1}\mathrm{P}\right|$

$\Rightarrow \mathrm{y}=\left|\left({\mathrm{P}}^{-1}\mathrm{I}\right)( \mathrm{A}-2\mathrm{I})\mathrm{P}\right|$

$\Rightarrow \mathrm{y}=\left|{\mathrm{P}}^{-1}\mathrm{I}\right||\mathrm{A}-2\mathrm{I}|\left|\mathrm{P}\right|$

We know that, $\left|\mathrm{A}\right|=\frac{1}{\left|{\mathrm{A}}^{-1}\right|}$.

$\Rightarrow \mathrm{y}=|\mathrm{A}-2\mathrm{I}|$

$\Rightarrow \mathrm{y}=\left|\begin{array}{ccc}2& 1& 2\\ 6& 2& 11\\ 3& 3& 2\end{array}\right|-\left|\begin{array}{lll}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right|$

$\Rightarrow \mathrm{y}=\left|\begin{array}{ccc}0& 1& 2\\ 6& 0& 11\\ 3& 3& 0\end{array}\right|$

$\Rightarrow \mathrm{y}=0-\left(0-33\right)+2\left(18-0\right)$

$\Rightarrow \mathrm{y}=69$

Now, prime factor of $69$ is $3$ and $23$.

So, sum $=26$