If `{[(5,1,4),(7,6,2),(1,3,5)][(1,6,-7),(6,2,4),(-7,4,3)][(5,7,1),(1,6,3),(4,2,5)]}^(2020)=[(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))]`, then the value of `2|a_(2)-b_(1)|+3|a_(3)-c_(1)|+4|b_(3)-c_(2)|` is equal to

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to analyze the given matrices and their properties. Let's break down the solution step by step.

### Step 1: Identify the matrices
We have three matrices:
- Matrix A: \((5, 1, 4), (7, 6, 2), (1, 3, 5)\)
- Matrix B: \((1, 6, -7), (6, 2, 4), (-7, 4, 3)\)
- Matrix C: \((5, 7, 1), (1, 6, 3), (4, 2, 5)\)

### Step 2: Multiply the matrices
We need to compute the product of these matrices and raise it to the power of 2020:
\[ X = (A \cdot B \cdot C)^{2020} \]

### Step 3: Analyze the symmetry
From the video transcript, we learn that matrix B is symmetric. A symmetric matrix has the property that its transpose is equal to itself:
\[ B = B^T \]

### Step 4: Use properties of transposes
We can take the transpose of the entire expression:
\[ X^T = (A \cdot B \cdot C)^{2020}^T = (C^T \cdot B^T \cdot A^T)^{2020} \]
Since \(B^T = B\), we have:
\[ X^T = (C^T \cdot B \cdot A^T)^{2020} \]

### Step 5: Establish equality
Given that \(X = X^T\), we conclude that \(X\) is symmetric. This implies:
\[ a_2 = b_1, \quad a_3 = c_1, \quad b_3 = c_2 \]

### Step 6: Calculate the expression
Now, we need to evaluate:
\[ 2|a_2 - b_1| + 3|a_3 - c_1| + 4|b_3 - c_2| \]

Since we established that:
- \(a_2 = b_1\)
- \(a_3 = c_1\)
- \(b_3 = c_2\)

This leads to:
\[ |a_2 - b_1| = 0, \quad |a_3 - c_1| = 0, \quad |b_3 - c_2| = 0 \]

### Step 7: Substitute values into the expression
Substituting these values into the expression:
\[ 2 \cdot 0 + 3 \cdot 0 + 4 \cdot 0 = 0 \]

### Final Answer
Thus, the value of \(2|a_2 - b_1| + 3|a_3 - c_1| + 4|b_3 - c_2|\) is:
\[ \boxed{0} \]