Matrices & Determinants
Similar Matrices — Determinant and Prime Factors
nta_pyq_2024_jan
Grade 12
Question:
Let $A=\begin{bmatrix}2&1&2\\6&2&11\\3&3&2\end{bmatrix}$ and $P=\begin{bmatrix}1&2&0\\5&0&2\\7&1&5\end{bmatrix}$. The sum of the prime factors of $|P^{-1}AP-2I|$ is equal to
Step-by-Step Solution
Key Concept: $|P^{-1}AP-2I|=|P^{-1}(A-2I)P|=|P^{-1}||A-2I||P|=|A-2I|$. Compute $A-2I$ and its determinant.
$|P^{-1}AP-2I|=|A-2I|=69$. Prime factors: $3$ and $23$. Sum $=26$.
Correct Answer: 1