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

Options

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

Answer

To solve 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 can follow these steps:

### Step 1: Understand the properties of \( \omega \)
The cube roots of unity satisfy the equations:
1. \( \omega^3 = 1 \)
2. \( 1 + \omega + \omega^2 = 0 \)

### Step 2: Write the determinant
We can express the determinant as:
\[ D = \begin{vmatrix} a & b\omega^2 & a\omega \\ b\omega & c & b\omega^2 \\ c\omega^2 & a\omega & c \end{vmatrix} \]

### Step 3: Apply the determinant properties
We can 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:
\[ \begin{vmatrix} c & b\omega^2 \\ a\omega & c \end{vmatrix} = c \cdot c - b\omega^2 \cdot a\omega = c^2 - ab\omega^3 = c^2 - ab \]
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:
\[ \begin{vmatrix} b\omega & c \\ c\omega^2 & a\omega \end{vmatrix} = b\omega \cdot a\omega - c \cdot c\omega^2 = ab\omega^2 - c^2\omega^2 = \omega^2(ab - c^2) \]

### Step 5: Substitute back into the determinant
Now substitute these back into the expression for \( D \):
\[ D = a(c^2 - ab) - b\omega^2 \cdot bc(\omega - 1) + a\omega \cdot \omega^2(ab - c^2) \]

### Step 6: Simplify the expression
1. The first term is \( ac^2 - a^2b \).
2. The second term becomes \( -b^2c\omega^2(\omega - 1) = -b^2c\omega^3 + b^2c\omega^2 = -b^2c + b^2c\omega^2 \).
3. The third term simplifies to \( a\omega^3(ab - c^2) = a(ab - c^2) \).

### Step 7: Combine all terms
Now combine all the terms:
\[ D = ac^2 - a^2b - b^2c + b^2c\omega^2 + a(ab - c^2) \]
Since \( \omega^3 = 1 \), we can simplify further:
\[ D = ac^2 - a^2b - b^2c + b^2c - ac^2 = 0 \]

### Final Result
Thus, the value of the determinant is:
\[ \boxed{0} \]