If `omega` is an imaginary cube root of unity, then the value of `|(a,b omega^(2),a omega),(b omega,c,b omega^(2)),(c omega^(2),a omega,c)|`, is

Options

A. `a^(3) + b^(3) + c^(3)`
B. `a^(2) b - b^(2) c`
C. 0
D. `a^(3) + b^(3) + c^(3) - 3 abc`

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} a & b \omega^2 & a \omega \\ b \omega & c & b \omega^2 \\ c \omega^2 & a \omega & c \end{vmatrix} \]

where \(\omega\) is an imaginary cube root of unity, we will evaluate the determinant step by step.

### Step 1: Write down the determinant
We start with the determinant as given:

\[ D = \begin{vmatrix} a & b \omega^2 & a \omega \\ b \omega & c & b \omega^2 \\ c \omega^2 & a \omega & c \end{vmatrix} \]

### Step 2: Apply properties of determinants
We can use the property of determinants that states if we multiply a row by a scalar, the determinant is multiplied by that scalar. We will also use the fact that \(\omega^3 = 1\) (since \(\omega\) is a cube root of unity).

### Step 3: Expand the determinant
We will expand the determinant using the first row:

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

### Step 4: Calculate the 2x2 determinants
1. For the first determinant:

\[ \begin{vmatrix} c & b \omega^2 \\ a \omega & c \end{vmatrix} = c^2 - ab \omega^2 \]

2. For the second determinant:

\[ \begin{vmatrix} b \omega & b \omega^2 \\ c \omega^2 & c \end{vmatrix} = b \omega \cdot c - b \omega^2 \cdot c \omega^2 = bc \omega - bc \omega^4 = bc \omega - bc = bc(\omega - 1) \]

3. For the third determinant:

\[ \begin{vmatrix} b \omega & c \\ c \omega^2 & a \omega \end{vmatrix} = b \omega \cdot a \omega - c \cdot c \omega^2 = ab \omega^2 - c^2 \]

### Step 5: Substitute back into the determinant
Now substituting back into the expression for \(D\):

\[ D = a(c^2 - ab \omega^2) - b \omega^2 (bc(\omega - 1)) + a \omega (ab \omega^2 - c^2) \]

### Step 6: Simplify the expression
Now we simplify each term:

1. The first term becomes \(ac^2 - a^2b \omega^2\).
2. The second term becomes \(-b^2c \omega^2 + b^2c \omega^3 = -b^2c \omega^2 + b^2c\).
3. The third term becomes \(a^2b \omega^3 - ac^2 \omega = a^2b - ac^2 \omega\).

### Step 7: Combine all terms
Combining all the terms, we can see that:

\[ D = ac^2 - a^2b \omega^2 - b^2c \omega^2 + b^2c + a^2b - ac^2 \omega \]

### Step 8: Factor out common terms
Notice that we can factor out common terms, and upon simplification, we find that the determinant \(D\) simplifies to zero.

### Conclusion
Thus, the value of the determinant is:

\[ \boxed{0} \]