If `omega ` is cube root of unity, then `| {:(1, omega, omega ^(2)),( omega, omega ^(2) , 1),( omega ^(2) , 1, omega):}|` is equal to

Options

A. 1
B. `omega`
C. `omega ^(2)`
D. 0

Answer

To solve the problem, we need to find the determinant of the following matrix:

\[ \begin{pmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{pmatrix} \]

where \(\omega\) is a cube root of unity. The cube roots of unity satisfy the equation \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\).

### Step 1: Write down the determinant

The determinant of a 3x3 matrix can be calculated using the formula:

\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

\[ |A| = 1 \cdot (\omega^2 \cdot \omega - 1 \cdot \omega^2) - \omega \cdot (\omega \cdot \omega - 1 \cdot \omega^2) + \omega^2 \cdot (\omega \cdot 1 - \omega^2 \cdot \omega) \]

### Step 2: Calculate each term

1. First term:
\[ 1 \cdot (\omega^2 \cdot \omega - 1 \cdot \omega^2) = \omega^3 - \omega^2 = 1 - \omega^2 \]

2. Second term:
\[ -\omega \cdot (\omega^2 - \omega^2) = -\omega \cdot 0 = 0 \]

3. Third term:
\[ \omega^2 \cdot (\omega - \omega^3) = \omega^2 \cdot (\omega - 1) = \omega^2 \cdot \omega - \omega^2 = \omega^3 - \omega^2 = 1 - \omega^2 \]

### Step 3: Combine the terms

Now, we combine the results from the three terms:

\[ |A| = (1 - \omega^2) + 0 + (1 - \omega^2) = 2 - 2\omega^2 \]

### Step 4: Use properties of cube roots of unity

Since \(1 + \omega + \omega^2 = 0\), we can express \(\omega^2\) as \(-1 - \omega\). Thus:

\[ |A| = 2 - 2(-1 - \omega) = 2 + 2 + 2\omega = 4 + 2\omega \]

However, we need to evaluate the determinant more carefully.

### Step 5: Recognize that the determinant is zero

Using the properties of cube roots of unity, we can see that the rows of the matrix are linearly dependent. Specifically, if we add all three rows, we get:

\[ (1 + \omega + \omega^2, \omega + \omega^2 + 1, \omega^2 + 1 + \omega) = (0, 0, 0) \]

This indicates that the determinant is zero.

### Final Answer

Thus, the value of the determinant is:

\[ \boxed{0} \]