If `omega` is a non-real cube root of unity, then `Delta = |(a_(1) + b_(1) omega,a_(1) omega^(2) + b_(1),a_(1) + b_(1) + c_(1) omega^(2)),(a_(2) + b_(2) omega,a_(2) omega^(2) + b_(2),a_(2) + b_(2) + c_(2) omega^(2)),(a_(3) + b_(3) omega,a_(3) omega^(2) + b_(3),a_(3) + b_(3) + c_(3) omega^(2))|` is equal to

Options

A. `-1`
B. 0
C. `-omega^(2)`
D. none of these

Explanation

To solve the problem, we need to evaluate the determinant given the matrix formed by the expressions involving the non-real cube root of unity, denoted as \( \omega \).

### Step-by-Step Solution:

1. **Understanding the Cube Root of Unity**:
- The non-real cube roots of unity are given by \( \omega = e^{2\pi i / 3} \) and \( \omega^2 = e^{4\pi i / 3} \).
- They satisfy the equation \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).

2. **Setting Up the Determinant**:
- The determinant \( \Delta \) is given by:
\[ \Delta = \begin{vmatrix} a_1 + b_1 \omega & a_1 \omega^2 + b_1 & a_1 + b_1 + c_1 \omega^2 \\ a_2 + b_2 \omega & a_2 \omega^2 + b_2 & a_2 + b_2 + c_2 \omega^2 \\ a_3 + b_3 \omega & a_3 \omega^2 + b_3 & a_3 + b_3 + c_3 \omega^2 \end{vmatrix} \]

3. **Column Operations**:
- We can multiply the first column by \( \omega^2 \):
\[ \text{First column becomes: } (a_1 \omega^2 + b_1, a_2 \omega^2 + b_2, a_3 \omega^2 + b_3) \]
- Since \( \omega^3 = 1 \), we can replace \( b_1 \omega^3 \) with \( b_1 \).

4. **Resulting Matrix**:
- The matrix now looks like:
\[ \begin{vmatrix} a_1 \omega^2 + b_1 & a_1 \omega^2 + b_1 & a_1 + b_1 + c_1 \omega^2 \\ a_2 \omega^2 + b_2 & a_2 \omega^2 + b_2 & a_2 + b_2 + c_2 \omega^2 \\ a_3 \omega^2 + b_3 & a_3 \omega^2 + b_3 & a_3 + b_3 + c_3 \omega^2 \end{vmatrix} \]

5. **Identical Columns**:
- Notice that the first two columns of the determinant are now identical:
\[ \begin{vmatrix} a_1 \omega^2 + b_1 & a_1 \omega^2 + b_1 & a_1 + b_1 + c_1 \omega^2 \\ a_2 \omega^2 + b_2 & a_2 \omega^2 + b_2 & a_2 + b_2 + c_2 \omega^2 \\ a_3 \omega^2 + b_3 & a_3 \omega^2 + b_3 & a_3 + b_3 + c_3 \omega^2 \end{vmatrix} \]

6. **Conclusion**:
- Since two columns of the determinant are identical, the value of the determinant is zero:
\[ \Delta = 0 \]

### Final Answer:
\[ \Delta = 0 \]