Let`A=[[2, 1, 2], [6, 2, 11], [3, 3, 2]]` and `P=[[1, 2, 0], [5, 0, 2], [7, 1, 5]]`. The sum of the prime factors of is equal to

Options

A. 26
B. 27
C. 66
D. 23

Answer

To solve the problem, we need to find the sum of the prime factors of a certain determinant derived from the matrices \( A \) and \( P \). Let's go through the steps systematically.

### Step 1: Define the matrices
We have the following matrices:
\[ A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}, \quad P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix} \]

### Step 2: Compute \( 2I \)
We need to compute \( 2I \), where \( I \) is the identity matrix of the same order as \( A \):
\[ 2I = 2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]

### Step 3: Compute \( A - 2I \)
Now, we subtract \( 2I \) from \( A \):
\[ A - 2I = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix} - \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 2 \\ 6 & 0 & 11 \\ 3 & 3 & 0 \end{bmatrix} \]

### Step 4: Compute the determinant of \( A - 2I \)
We need to find the determinant of the matrix \( A - 2I \):
\[ \text{det}(A - 2I) = \begin{vmatrix} 0 & 1 & 2 \\ 6 & 0 & 11 \\ 3 & 3 & 0 \end{vmatrix} \]

Using the determinant formula for a \( 3 \times 3 \) matrix:
\[ \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the elements of the second and third rows respectively.

Calculating the determinant:
\[ = 0 \cdot (0 \cdot 0 - 11 \cdot 3) - 1 \cdot (6 \cdot 0 - 11 \cdot 3) + 2 \cdot (6 \cdot 3 - 0 \cdot 3) \]
\[ = 0 - 1 \cdot (-33) + 2 \cdot 18 \]
\[ = 33 + 36 = 69 \]

### Step 5: Find the prime factors of 69
Now, we need to find the prime factors of 69. The prime factorization of 69 is:
\[ 69 = 3 \times 23 \]

### Step 6: Sum the prime factors
Now, we sum the prime factors:
\[ 3 + 23 = 26 \]

### Final Result
Thus, the sum of the prime factors of the determinant \( \text{det}(A - 2I) \) is:
\[ \boxed{26} \]