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Let's analyze the question in detail:

### Given Information:
- \( M = \begin{bmatrix} \omega_1 & \omega_2 & \omega_3 \\ \omega_4 & \omega_5 & \omega_6 \\ \omega_7 & \omega_8 & \omega_9 \end{bmatrix} \) is a \( 3 \times 3 \) invertible matrix.
- \( \omega_1 = \omega_4 = \omega_7 = \omega^2 \), \( \omega_2 = \omega_5 = \omega_8 = \omega \), and \( \omega_3 = \omega_6 = \omega_9 = 1 \).
- \( \omega \neq 1 \) and is a cube root of unity.
- \( \omega_1, \omega_2, \omega_3 \in \{\omega, \omega^2, 1\} \).

The question is asking to determine how many such matrices \( M \) are possible.

### Cube Root of Unity Property:
The cube roots of unity are:
- \( 1 \)
- \( \omega \), where \( \omega \) is a complex number such that \( \omega^3 = 1 \)
- \( \omega^2 \)

The matrix is given in a fixed form:
- The diagonal elements are fixed as \( \omega^2 \) for the first column, \( \omega \) for the second column, and \( 1 \) for the third column.

### Invertibility Condition:
To ensure \( M \) is invertible, the determinant of \( M \) must be non-zero.

Since the matrix structure is fixed by the values of \( \omega \), \( \omega^2 \), and 1 in each position, and these values already satisfy the non-singularity condition, the matrix will have a non-zero determinant. Therefore, there is only **1 such matrix possible**.

### Final Answer:
The number of possible matrices \( M \) is **1**.

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Do you have any questions or need further clarification?

### Related Questions:
1. What is the general condition for a matrix to be invertible?
2. Can a matrix with all elements as roots of unity be invertible?
3. How is the determinant of a matrix related to its invertibility?
4. What are the cube roots of unity and their properties?
5. How would the answer change if \( \omega \) were not a cube root of unity?

### Tip:
When working with cube roots of unity, remember that \( \omega^3 = 1 \), and this property simplifies many calculations involving powers of \( \omega \).

Explore how to determine the number of possible invertible matrices using cube roots of unity. Understand the fixed structure of the matrix and the conditions under which it remains invertible. Suitable for undergraduate level study in linear algebra.

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