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
Solution
<p>$$\begin{aligned}
\left|\mathrm{P}^{-1} \mathrm{AP}-2 \mathrm{I}\right| & =\left|\mathrm{P}^{-1} \mathrm{AP}-2 \mathrm{P}^{-1} \mathrm{P}\right| \\
& =\left|\mathrm{P}^{-1}(\mathrm{~A}-2 \mathrm{I}) \mathrm{P}\right| \\
& =\left|\mathrm{P}^{-1}\right||\mathrm{A}-2 \mathrm{I}||\mathrm{P}| \\
& =|\mathrm{A}-2 \mathrm{I}| \\
& =\left|\begin{array}{ccc}
0 & 1 & 2 \\
6 & 0 & 11 \\
3 & 3 & 0
\end{array}\right|=69
\end{aligned}$$</p>
<p>So, Prime factor of 69 is 3 & 23</p>
<p>So, sum = 26</p>
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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