If $\omega$ is the cube root of unity, then the inverse of $A=\begin{bmatrix}1&1&1\ 1&\omega&\omega^{2}\ 1&\omega^{2}&\omega\end{bmatrix}$ is :

Question

Accepted Answer

To determine the inverse of the matrix $ A = \begin{bmatrix} 1 & 1 & 1 \ 1 & \omega & \omega^2 \ 1 & \omega^2 & \omega \end{bmatrix} $, where $\omega$ is a cube root of unity, we need to understand the properties of cube roots of unity and how they affect matrix operations.

### Properties of Cube Roots of Unity:
1. $\omega^3 = 1$
2. $1 + \omega + \omega^2 = 0$

### Explanation for Option 1:
The matrix $ A $ is a Vandermonde matrix, which has a well-known inverse when the elements are distinct roots of unity. The inverse of a Vandermonde matrix with cube roots of unity is given by:

$$
A^{-1} = \frac{1}{n} \begin{bmatrix} 1 & 1 & 1 \ 1 & \omega^2 & \omega \ 1 & \omega & \omega^2 \end{bmatrix}
$$

where $ n $ is the order of the roots, which is 3 in this case. Therefore, the inverse is:

$$
A^{-1} = \frac{1}{3} \begin{bmatrix} 1 & 1 & 1 \ 1 & \omega^2 & \omega \ 1 & \omega & \omega^2 \end{bmatrix}
$$

This matches Option 1, confirming it as the correct answer.

### Explanation for Other Options:

- **Option 2: $\frac{1}{4}\begin{bmatrix}1&1&1\ 1&\omega&\omega^{2}\ 1&\omega^{2}&\omega\end{bmatrix}$**
  - The factor $\frac{1}{4}$ is incorrect because the determinant of the matrix $ A $ is 3, not 4. The correct factor should be $\frac{1}{3}$.

- **Option 3: $\frac{1}{2}\begin{bmatrix}1&1&1\ 1&\omega^{2}&\omega\ 1&\omega&\omega^{2}\end{ matrix}$**
  - The factor $\frac{1}{2}$ is incorrect for the same reason as Option 2. The determinant of $ A $ is 3, so the factor must be $\frac{1}{3}$.

- **Option 4: $\begin{bmatrix}1&1&1\ 1&\omega^{2}&\omega\ 1&\omega&\omega^{2}\end{bmatrix}$**
  - This option does not include any scalar factor, which is necessary to account for the determinant of the original matrix. The absence of the factor $\frac{1}{3}$ makes this option incorrect.

In conclusion, Option 1 is the correct answer because it accurately reflects the inverse of the matrix $ A $ with the appropriate scalar factor derived from the determinant of the matrix.