With `1,omega,omega^(2)` as cube roots of unity, inverse of which of the following matrices exists?

Options

A. `[{:(1,omega),(omega,omega):}]`
B. `[{:(omega^(2),1),(1,omega):}]`
C. `[{:(omega,omega^(2)),(omega^(2),1):}]`
D. None of these

Answer

To determine the existence of the inverse of the given matrices with cube roots of unity \(1, \omega, \omega^2\), we need to calculate the determinant of each matrix. The inverse of a matrix exists if and only if its determinant is non-zero.

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

### Step 2: Calculate the determinant of each matrix

**Matrix 1:**
\[ A_1 = \begin{bmatrix} 1 & \omega \\ \omega & \omega^2 \end{bmatrix} \]
The determinant is calculated as:
\[ \text{det}(A_1) = (1)(\omega^2) - (\omega)(\omega) = \omega^2 - \omega^2 = 0 \]

**Matrix 2:**
\[ A_2 = \begin{bmatrix} \omega^2 & 1 \\ 1 & \omega \end{bmatrix} \]
The determinant is calculated as:
\[ \text{det}(A_2) = (\omega^2)(\omega) - (1)(1) = \omega^3 - 1 = 1 - 1 = 0 \]

**Matrix 3:**
\[ A_3 = \begin{bmatrix} \omega & \omega^2 \\ \omega^2 & 1 \end{bmatrix} \]
The determinant is calculated as:
\[ \text{det}(A_3) = (\omega)(1) - (\omega^2)(\omega^2) = \omega - \omega^4 \]
Since \(\omega^4 = \omega\) (because \(\omega^3 = 1\)), we have:
\[ \text{det}(A_3) = \omega - \omega = 0 \]

### Step 3: Conclusion
Since the determinants of all three matrices are zero, none of the matrices have an inverse.

### Final Answer
The inverse of none of the matrices exists.

---