If `omega` is a complex cube root of unity, then value of
`Delta=|(a_(1)+b_(1)omega, a_(1)omega^(2)+b_(1),c_(1)+b_(1)omega),(a_(2)+b_(2)omega,a_(2)omega^(2)+b_(2),c_(2)+b_(2)omega),(a_(3)+b_(3)omega,a_(3)omega^(2)+b_(3),c_(3)+b_(3)omega)|` is

Options

A. 0
B. `-1`
C. 2
D. None of these

Answer

To find the value of the determinant \( \Delta \) given by

\[ \Delta = \begin{vmatrix} a_1 + b_1 \omega & a_1 \omega^2 + b_1 & c_1 + b_1 \omega \\ a_2 + b_2 \omega & a_2 \omega^2 + b_2 & c_2 + b_2 \omega \\ a_3 + b_3 \omega & a_3 \omega^2 + b_3 & c_3 + b_3 \omega \end{vmatrix} \]

where \( \omega \) is a complex cube root of unity, we can proceed with the following steps:

### Step 1: Understand the properties of \( \omega \)
Recall that \( \omega \) satisfies the equations:
1. \( 1 + \omega + \omega^2 = 0 \)
2. \( \omega^3 = 1 \)

### Step 2: Manipulate the second column
We will multiply the second column by \( \omega \):
\[ \text{Column 2} = \omega \cdot (a_1 \omega^2 + b_1, a_2 \omega^2 + b_2, a_3 \omega^2 + b_3) \]
This gives us:
\[ \Delta = \begin{vmatrix} a_1 + b_1 \omega & \omega(a_1 \omega^2 + b_1) & c_1 + b_1 \omega \\ a_2 + b_2 \omega & \omega(a_2 \omega^2 + b_2) & c_2 + b_2 \omega \\ a_3 + b_3 \omega & \omega(a_3 \omega^2 + b_3) & c_3 + b_3 \omega \end{vmatrix} \]

### Step 3: Simplify the second column
Since \( \omega^3 = 1 \), we can simplify:
\[ \omega(a_i \omega^2 + b_i) = a_i + b_i \omega \]
Thus, the determinant becomes:
\[ \Delta = \begin{vmatrix} a_1 + b_1 \omega & a_1 + b_1 \omega & c_1 + b_1 \omega \\ a_2 + b_2 \omega & a_2 + b_2 \omega & c_2 + b_2 \omega \\ a_3 + b_3 \omega & a_3 + b_3 \omega & c_3 + b_3 \omega \end{vmatrix} \]

### Step 4: Identify proportional columns
Now we see that the first and second columns are identical:
\[ \text{Column 1} = \text{Column 2} \]
This means that the columns of the determinant are linearly dependent.

### Step 5: Conclude the value of the determinant
Since the determinant has two identical columns, we conclude that:
\[ \Delta = 0 \]

Thus, the value of the determinant \( \Delta \) is \( \boxed{0} \).

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