The value of `|{:(-1,,2,,1),(3+2sqrt(2),,2+2sqrt(2),,1),(3-2sqrt(2),,2-2sqrt(2),,1):}|` is equal to

Options

A. `0`
B. `-16sqrt(2)`
C. `-8sqrt(2)`
D. none of these

Explanation

To find the value of the determinant \( | \begin{pmatrix} -1 & 2 & 1 \\ 3 + 2\sqrt{2} & 2 + 2\sqrt{2} & 1 \\ 3 - 2\sqrt{2} & 2 - 2\sqrt{2} & 1 \end{pmatrix} | \), we will follow these steps:

### Step 1: Write the Determinant
We start by writing the determinant as follows:
\[ D = \begin{vmatrix} -1 & 2 & 1 \\ 3 + 2\sqrt{2} & 2 + 2\sqrt{2} & 1 \\ 3 - 2\sqrt{2} & 2 - 2\sqrt{2} & 1 \end{vmatrix} \]

### Step 2: Apply Row Transformations
We will perform row transformations to simplify the determinant. Specifically, we will perform the following operations:
- \( R_1 \rightarrow R_1 - R_2 \)
- \( R_2 \rightarrow R_2 - R_3 \)

After performing these operations, we will have:
\[ D = \begin{vmatrix} -4 - 2\sqrt{2} & 0 & 0 \\ 0 & 4\sqrt{2} & 0 \\ 3 - 2\sqrt{2} & 2 - 2\sqrt{2} & 1 \end{vmatrix} \]

### Step 3: Expand the Determinant
Next, we will expand the determinant along the third column:
\[ D = 0 - 0 + 1 \cdot \begin{vmatrix} -4 - 2\sqrt{2} & 0 \\ 3 - 2\sqrt{2} & 2 - 2\sqrt{2} \end{vmatrix} \]

### Step 4: Calculate the 2x2 Determinant
Now we need to calculate the determinant of the 2x2 matrix:
\[ \begin{vmatrix} -4 - 2\sqrt{2} & 0 \\ 3 - 2\sqrt{2} & 2 - 2\sqrt{2} \end{vmatrix} = (-4 - 2\sqrt{2})(2 - 2\sqrt{2}) - (0)(3 - 2\sqrt{2}) \]
This simplifies to:
\[ = (-4 - 2\sqrt{2})(2 - 2\sqrt{2}) = -8 + 8\sqrt{2} - 4\sqrt{2} + 4 = -4 + 4\sqrt{2} \]

### Step 5: Substitute Back into the Determinant
Now substituting back, we have:
\[ D = 1 \cdot (-4 + 4\sqrt{2}) = -4 + 4\sqrt{2} \]

### Final Answer
Thus, the value of the determinant is:
\[ D = -4 + 4\sqrt{2} \]