Matrices Question 7
Question 7 - 2024 (29 Jan Shift 2)
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 $\left|P^{-1} AP-2 I\right|$ is equal to
(1) 26
(2) 27
(3) 66
(4) 23
Show Answer
Answer (1)
Solution
$$ \begin{aligned} \left|P^{-1} AP-2 I\right| & =\left|P^{-1} AP-2 P^{-1} P\right| \\ & =\left|P^{-1}(A-2 I) P\right| \\ & =\left|P^{-1}\right||A-2 I||P| \\ & =|A-2 I| \\ & =\left|\begin{array}{ccc} 0 & 1 & 2 \\ 6 & 0 & 11 \\ 3 & 3 & 0 \end{array}\right|=69 \end{aligned} $$
So, Prime factor of 69 is $3 \& 23$
So, sum $=26$
