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

### Step 1: Identify the matrices
We have three matrices:
- Matrix A:
\[ A = \begin{pmatrix} 5 & 1 & 4 \\ 7 & 6 & 2 \\ 1 & 3 & 5 \end{pmatrix} \]
- Matrix B:
\[ B = \begin{pmatrix} 1 & 6 & -7 \\ 6 & 2 & 4 \\ -7 & 4 & 3 \end{pmatrix} \]
- Matrix C (which is the transpose of B):
\[ B^T = \begin{pmatrix} 1 & 6 & -7 \\ 6 & 2 & 4 \\ -7 & 4 & 3 \end{pmatrix} \]

### Step 2: Compute the product of the matrices
We need to compute the product of these matrices raised to the power of 2020:
\[ X = (A \cdot B \cdot B^T)^{2020} \]

### Step 3: Properties of symmetric matrices
Since matrix B is symmetric (i.e., \(B = B^T\)), we can use the property of symmetric matrices:
- The transpose of a product of matrices is the product of their transposes in reverse order.

### Step 4: Simplifying the expression
Using the property of symmetric matrices, we can conclude that:
\[ X = (A \cdot B \cdot B)^{2020} \]
This means that \(X\) is also symmetric.

### Step 5: Identifying relationships between elements
Since \(X\) is symmetric, we can establish the following relationships:
- \(a_2 = b_1\)
- \(a_3 = c_1\)
- \(b_3 = c_2\)

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

### Step 7: Evaluate the expression
Since \(a_2 = b_1\), \(a_3 = c_1\), and \(b_3 = c_2\), we have:
- \(2|a_2 - b_1| = 2|0| = 0\)
- \(3|a_3 - c_1| = 3|0| = 0\)
- \(4|b_3 - c_2| = 4|0| = 0\)

Adding these together gives:
\[ 0 + 0 + 0 = 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]