With the numbers `2, 4, 6, 8,` all the possible determinants with these four different elements are constructed. What is the sum of the values of all such determinants?

Options

A. `128`
B. `64`
C. `32`
D. `0`

Answer

To find the sum of the values of all possible determinants constructed with the numbers 2, 4, 6, and 8, we will follow these steps:

### Step 1: Understand the Determinant of a 2x2 Matrix
The determinant of a 2x2 matrix is calculated using the formula:
\[ \text{det} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc \]
where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix.

### Step 2: List All Possible 2x2 Matrices
We will create all possible 2x2 matrices using the numbers 2, 4, 6, and 8. The elements must be different in each matrix.

The possible matrices are:
1. \(\begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}\)
2. \(\begin{pmatrix} 2 & 6 \\ 4 & 8 \end{pmatrix}\)
3. \(\begin{pmatrix} 2 & 8 \\ 4 & 6 \end{pmatrix}\)
4. \(\begin{pmatrix} 4 & 2 \\ 6 & 8 \end{pmatrix}\)
5. \(\begin{pmatrix} 4 & 6 \\ 2 & 8 \end{pmatrix}\)
6. \(\begin{pmatrix} 4 & 8 \\ 2 & 6 \end{pmatrix}\)
7. \(\begin{pmatrix} 6 & 2 \\ 4 & 8 \end{pmatrix}\)
8. \(\begin{pmatrix} 6 & 4 \\ 2 & 8 \end{pmatrix}\)
9. \(\begin{pmatrix} 6 & 8 \\ 2 & 4 \end{pmatrix}\)
10. \(\begin{pmatrix} 8 & 2 \\ 4 & 6 \end{pmatrix}\)
11. \(\begin{pmatrix} 8 & 4 \\ 2 & 6 \end{pmatrix}\)
12. \(\begin{pmatrix} 8 & 6 \\ 2 & 4 \end{pmatrix}\)

### Step 3: Calculate the Determinant for Each Matrix
Now we will calculate the determinant for each of the matrices listed above.

1. \(\text{det} \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix} = (2)(8) - (4)(6) = 16 - 24 = -8\)
2. \(\text{det} \begin{pmatrix} 2 & 6 \\ 4 & 8 \end{pmatrix} = (2)(8) - (6)(4) = 16 - 24 = -8\)
3. \(\text{det} \begin{pmatrix} 2 & 8 \\ 4 & 6 \end{pmatrix} = (2)(6) - (8)(4) = 12 - 32 = -20\)
4. \(\text{det} \begin{pmatrix} 4 & 2 \\ 6 & 8 \end{pmatrix} = (4)(8) - (2)(6) = 32 - 12 = 20\)
5. \(\text{det} \begin{pmatrix} 4 & 6 \\ 2 & 8 \end{pmatrix} = (4)(8) - (6)(2) = 32 - 12 = 20\)
6. \(\text{det} \begin{pmatrix} 4 & 8 \\ 2 & 6 \end{pmatrix} = (4)(6) - (8)(2) = 24 - 16 = 8\)
7. \(\text{det} \begin{pmatrix} 6 & 2 \\ 4 & 8 \end{pmatrix} = (6)(8) - (2)(4) = 48 - 8 = 40\)
8. \(\text{det} \begin{pmatrix} 6 & 4 \\ 2 & 8 \end{pmatrix} = (6)(8) - (4)(2) = 48 - 8 = 40\)
9. \(\text{det} \begin{pmatrix} 6 & 8 \\ 2 & 4 \end{pmatrix} = (6)(4) - (8)(2) = 24 - 16 = 8\)
10. \(\text{det} \begin{pmatrix} 8 & 2 \\ 4 & 6 \end{pmatrix} = (8)(6) - (2)(4) = 48 - 8 = 40\)
11. \(\text{det} \begin{pmatrix} 8 & 4 \\ 2 & 6 \end{pmatrix} = (8)(6) - (4)(2) = 48 - 8 = 40\)
12. \(\text{det} \begin{pmatrix} 8 & 6 \\ 2 & 4 \end{pmatrix} = (8)(4) - (6)(2) = 32 - 12 = 20\)

### Step 4: Sum All the Determinants
Now we will sum all the calculated determinants:
\[ \text{Sum} = (-8) + (-8) + (-20) + 20 + 20 + 8 + 40 + 40 + 8 + 40 + 40 + 20 \]
Calculating this gives:
\[ \text{Sum} = 0 \]

### Final Answer
The sum of the values of all such determinants is \(0\).