The determinant
`|(b_(1)+c_(1),c_(1)+a_(1),a_(1)+b_(1)),(b_(2)+c_(2),c_(2)+a_(2),a_(2)+b_(2)),(b_(3)+c_(3),c_(3)+a_(3),a_(3)+b_(3))|`

Options

A. `[{:(,a_(1),b_(1),c_(1)),(,a_(2),b_(2),c_(2)),(,a_(3),b_(3),c_(3)):}]`
B. `2[{:(,a_(1),b_(1),c_(1)),(,a_(2),b_(2),c_(2)),(,a_(3),b_(3),c_(3)):}]`
C. `3[{:(,a_(1),b_(1),c_(1)),(,a_(2),b_(2),c_(2)),(,a_(3),b_(3),c_(3)):}]`
D. none of these

Answer

To find the determinant

\[ D = \begin{vmatrix} b_1 + c_1 & c_1 + a_1 & a_1 + b_1 \\ b_2 + c_2 & c_2 + a_2 & a_2 + b_2 \\ b_3 + c_3 & c_3 + a_3 & a_3 + b_3 \end{vmatrix} \]

we will perform a series of column operations to simplify the determinant.

### Step 1: Subtract Column 2 from Column 1
We start by subtracting the second column from the first column:

\[ D = \begin{vmatrix} (b_1 + c_1) - (c_1 + a_1) & c_1 + a_1 & a_1 + b_1 \\ (b_2 + c_2) - (c_2 + a_2) & c_2 + a_2 & a_2 + b_2 \\ (b_3 + c_3) - (c_3 + a_3) & c_3 + a_3 & a_3 + b_3 \end{vmatrix} \]

This simplifies to:

\[ D = \begin{vmatrix} b_1 - a_1 & c_1 + a_1 & a_1 + b_1 \\ b_2 - a_2 & c_2 + a_2 & a_2 + b_2 \\ b_3 - a_3 & c_3 + a_3 & a_3 + b_3 \end{vmatrix} \]

### Step 2: Subtract Column 3 from Column 2
Next, we subtract the third column from the second column:

\[ D = \begin{vmatrix} b_1 - a_1 & (c_1 + a_1) - (a_1 + b_1) & a_1 + b_1 \\ b_2 - a_2 & (c_2 + a_2) - (a_2 + b_2) & a_2 + b_2 \\ b_3 - a_3 & (c_3 + a_3) - (a_3 + b_3) & a_3 + b_3 \end{vmatrix} \]

This simplifies to:

\[ D = \begin{vmatrix} b_1 - a_1 & c_1 - b_1 & a_1 + b_1 \\ b_2 - a_2 & c_2 - b_2 & a_2 + b_2 \\ b_3 - a_3 & c_3 - b_3 & a_3 + b_3 \end{vmatrix} \]

### Step 3: Add Column 1 and Column 2
Next, we add the first column to the second column:

\[ D = \begin{vmatrix} b_1 - a_1 & (b_1 - a_1) + (c_1 - b_1) & a_1 + b_1 \\ b_2 - a_2 & (b_2 - a_2) + (c_2 - b_2) & a_2 + b_2 \\ b_3 - a_3 & (b_3 - a_3) + (c_3 - b_3) & a_3 + b_3 \end{vmatrix} \]

This simplifies to:

\[ D = \begin{vmatrix} b_1 - a_1 & c_1 - a_1 & a_1 + b_1 \\ b_2 - a_2 & c_2 - a_2 & a_2 + b_2 \\ b_3 - a_3 & c_3 - a_3 & a_3 + b_3 \end{vmatrix} \]

### Step 4: Factor Out Common Terms
We can factor out 2 from the first column:

\[ D = 2 \begin{vmatrix} b_1 - a_1 & c_1 - a_1 & a_1 + b_1 \\ b_2 - a_2 & c_2 - a_2 & a_2 + b_2 \\ b_3 - a_3 & c_3 - a_3 & a_3 + b_3 \end{vmatrix} \]

### Step 5: Final Determinant
The determinant can be expressed as:

\[ D = 2 \begin{vmatrix} b_1 & c_1 & a_1 \\ b_2 & c_2 & a_2 \\ b_3 & c_3 & a_3 \end{vmatrix} \]

Thus, the final result is:

\[ D = 2 \cdot \text{det}(B) \]

Where \( B \) is the matrix formed by \( b_i, c_i, a_i \).