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Mathematics Question on Trigonometry

Let A=[212 6211 332] and P=[120 502 715].\text{Let } A = \begin{bmatrix} 2 & 1 & 2 \\\ 6 & 2 & 11 \\\ 3 & 3 & 2 \end{bmatrix} \text{ and } P = \begin{bmatrix} 1 & 2 & 0 \\\ 5 & 0 & 2 \\\ 7 & 1 & 5 \end{bmatrix}.The sum of the prime factors of ∣P−1AP−2I∣|P^{-1}AP - 2I| is equal to.

A

26

B

27

C

66

D

23

Answer

26

Explanation

Solution

Solution: We need to find ∣P−1AP−2I∣|P^{-1}AP - 2I|.

Step 1. Calculating ∣P−1AP−2I∣:|P^{-1}AP - 2I|:

∣P−1AP−2I∣=∣P−1AP−2P−1P∣|P^{-1}AP - 2I| = |P^{-1}AP - 2P^{-1}P|
=∣P−1(A−2I)P∣= |P^{-1}(A - 2I)P|
=∣P−1∣⋅∣A−2I∣⋅∣P∣= |P^{-1}| \cdot |A - 2I| \cdot |P|
=∣A−2I∣= |A - 2I|

Step 2. Calculating ∣A−2I∣:|A - 2I|:
A−2I=[212 6211 332]−[200 020 002]=[012 6011 330]A - 2I = \begin{bmatrix} 2 & 1 & 2 \\\ 6 & 2 & 11 \\\ 3 & 3 & 2 \end{bmatrix} - \begin{bmatrix} 2 & 0 & 0 \\\ 0 & 2 & 0 \\\ 0 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 2 \\\ 6 & 0 & 11 \\\ 3 & 3 & 0 \end{bmatrix}
∣A−2I∣=69|A - 2I| = 69

Step 3. The prime factors of 69 are 3 and 23, so the sum of the prime factors is:
3+23=263 + 23 = 26
The Correct Answer is: 26