If `omega` is a complex cube root of unity then the matrix `A = [(1, omega^(2),omega),(omega^(2),omega,1),(omega,1,omega^(2))]` is a

Options

A. singular matrix
B. non -singular matrix
C. skew symmetric matrix
D. none of these

Answer

To determine the properties of the matrix \( A \) given by

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

where \( \omega \) is a complex cube root of unity, we will first calculate the determinant of the matrix \( A \).

### Step 1: Recall the properties of \( \omega \)
The complex cube roots of unity are given by:
\[ \omega = e^{2\pi i / 3} \quad \text{and} \quad \omega^2 = e^{-2\pi i / 3} \]
These roots satisfy the equation:
\[ 1 + \omega + \omega^2 = 0 \]
Also, we have:
\[ \omega^3 = 1 \quad \text{and} \quad \omega^2 = \frac{1}{\omega} \]

### Step 2: Calculate the determinant of \( A \)
To find the determinant of \( A \), we can use the formula for the determinant of a \( 3 \times 3 \) matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix \( A \):
- \( a = 1, b = \omega^2, c = \omega \)
- \( d = \omega^2, e = \omega, f = 1 \)
- \( g = \omega, h = 1, i = \omega^2 \)

Substituting these values into the determinant formula:
\[ \text{det}(A) = 1(\omega \cdot \omega^2 - 1 \cdot 1) - \omega^2(\omega^2 \cdot \omega^2 - 1 \cdot \omega) + \omega(\omega^2 \cdot 1 - \omega \cdot \omega) \]

Calculating each term:
1. \( \omega \cdot \omega^2 = \omega^3 = 1 \)
2. \( 1 \cdot 1 = 1 \)
3. \( \omega^2 \cdot \omega^2 = \omega^4 = \omega \)
4. \( 1 \cdot \omega = \omega \)
5. \( \omega^2 \cdot 1 = \omega^2 \)
6. \( \omega \cdot \omega = \omega^2 \)

Substituting these results:
\[ \text{det}(A) = 1(1 - 1) - \omega^2(\omega - \omega) + \omega(\omega^2 - \omega^2) \]
\[ = 1(0) - \omega^2(0) + \omega(0) = 0 \]

### Conclusion
Since the determinant of the matrix \( A \) is \( 0 \), the matrix \( A \) is singular.