If `omega` is an imaginary cube root of unity, then the value of `|(1,omega^(2),1-omega^(4)),(omega,1,1+omega^(5)),(1,omega,omega^(2))|` is

Options

A. `-4`
B. `omega^(2)-4`
C. `omega^(2)`
D. 4

Answer

To solve the given problem, we need to evaluate the determinant:

\[ D = \begin{vmatrix} 1 & \omega^2 & 1 - \omega^4 \\ \omega & 1 & 1 + \omega^5 \\ 1 & \omega & \omega^2 \end{vmatrix} \]

### Step 1: Simplify the powers of \(\omega\)

Given that \(\omega\) is a cube root of unity, we have the following properties:
- \(\omega^3 = 1\)
- \(1 + \omega + \omega^2 = 0\)

Using these properties, we can simplify \(\omega^4\) and \(\omega^5\):
- \(\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega\)
- \(\omega^5 = \omega^3 \cdot \omega^2 = 1 \cdot \omega^2 = \omega^2\)

Now we can rewrite the determinant with these simplifications:

\[ D = \begin{vmatrix} 1 & \omega^2 & 1 - \omega \\ \omega & 1 & 1 + \omega^2 \\ 1 & \omega & \omega^2 \end{vmatrix} \]

### Step 2: Calculate the determinant

We can now calculate the determinant \(D\) using cofactor expansion along the first row:

\[ D = 1 \cdot \begin{vmatrix} 1 & 1 + \omega^2 \\ \omega & \omega^2 \end{vmatrix} - \omega^2 \cdot \begin{vmatrix} \omega & 1 + \omega^2 \\ 1 & \omega^2 \end{vmatrix} + (1 - \omega) \cdot \begin{vmatrix} \omega & 1 \\ 1 & \omega \end{vmatrix} \]

### Step 3: Evaluate the 2x2 determinants

1. **First determinant**:
\[ \begin{vmatrix} 1 & 1 + \omega^2 \\ \omega & \omega^2 \end{vmatrix} = 1 \cdot \omega^2 - \omega \cdot (1 + \omega^2) = \omega^2 - \omega - \omega^3 = \omega^2 - \omega - 1 \]

2. **Second determinant**:
\[ \begin{vmatrix} \omega & 1 + \omega^2 \\ 1 & \omega^2 \end{vmatrix} = \omega \cdot \omega^2 - 1 \cdot (1 + \omega^2) = \omega^3 - (1 + \omega^2) = 1 - 1 - \omega^2 = -\omega^2 \]

3. **Third determinant**:
\[ \begin{vmatrix} \omega & 1 \\ 1 & \omega \end{vmatrix} = \omega^2 - 1 \]

### Step 4: Substitute back into the determinant \(D\)

Now substituting back into the expression for \(D\):

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

Expanding this:

\[ D = \omega^2 - \omega - 1 + \omega^4 + (1 - \omega)(\omega^2 - 1) \]

Since \(\omega^4 = \omega\):

\[ D = \omega^2 - \omega - 1 + \omega + (1 - \omega)(\omega^2 - 1) \]

### Step 5: Simplify further

Now we simplify \(D\):

\[ D = \omega^2 - 1 + (1 - \omega)(\omega^2 - 1) \]

Expanding \((1 - \omega)(\omega^2 - 1)\):

\[ = \omega^2 - 1 - \omega^3 + \omega = \omega^2 - 1 - 1 + \omega = \omega^2 + \omega - 2 \]

So we have:

\[ D = \omega^2 - 1 + \omega^2 + \omega - 2 = 2\omega^2 + \omega - 3 \]

### Final Step: Evaluate \(D\)

Using the property \(1 + \omega + \omega^2 = 0\) to substitute for \(\omega\):

\[ D = 2\omega^2 + (-1 - \omega^2) - 3 = \omega^2 - 4 \]

Thus, the final value of the determinant is:

\[ \boxed{\omega^2 - 4} \]