Let `omega!=1` be cube root of unity and `S` be the set of all non-singular matrices of the form `[1a bomega1comega^2omega1],w h e r e` each of `a ,b ,a n dc` is either `omegaoromega^2dot` Then the number of distinct matrices in the set `S` is a. 2 b. `6` c. `4` d. `8`

Options

A. 2
B. 6
C. 4
D. 8

Explanation

To solve the problem, we need to determine the number of distinct non-singular matrices of the form:

\[ \begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix} \]

where \( a, b, c \) can each be either \( \omega \) or \( \omega^2 \), and \( \omega \) is a cube root of unity (i.e., \( \omega^3 = 1 \) and \( \omega \neq 1 \)).

### Step 1: Calculate the Determinant
The determinant of the matrix can be computed using the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = 1 \cdot \begin{vmatrix} 1 & c \\ \omega & 1 \end{vmatrix} - a \cdot \begin{vmatrix} \omega & c \\ \omega^2 & 1 \end{vmatrix} + b \cdot \begin{vmatrix} \omega & 1 \\ \omega^2 & \omega \end{vmatrix} \]

Calculating these 2x2 determinants:

1. \( \begin{vmatrix} 1 & c \\ \omega & 1 \end{vmatrix} = 1 \cdot 1 - c \cdot \omega = 1 - c\omega \)

2. \( \begin{vmatrix} \omega & c \\ \omega^2 & 1 \end{vmatrix} = \omega \cdot 1 - c \cdot \omega^2 = \omega - c\omega^2 \)

3. \( \begin{vmatrix} \omega & 1 \\ \omega^2 & \omega \end{vmatrix} = \omega \cdot \omega - 1 \cdot \omega^2 = \omega^2 - \omega^2 = 0 \)

Putting it all together, we have:

\[ \text{det}(A) = 1 - c\omega - a(\omega - c\omega^2) + b \cdot 0 \]

This simplifies to:

\[ \text{det}(A) = 1 - c\omega - a\omega + ac\omega^2 \]

### Step 2: Set the Determinant Not Equal to Zero
For the matrix to be non-singular, we require:

\[ 1 - c\omega - a\omega + ac\omega^2 \neq 0 \]

### Step 3: Analyze Possible Values for \( a, b, c \)
Each of \( a, b, c \) can be either \( \omega \) or \( \omega^2 \). Therefore, we have:

- \( a \) can take 2 values: \( \omega \) or \( \omega^2 \)
- \( b \) can take 2 values: \( \omega \) or \( \omega^2 \)
- \( c \) can take 2 values: \( \omega \) or \( \omega^2 \)

This gives us a total of \( 2 \times 2 \times 2 = 8 \) combinations of \( (a, b, c) \).

### Step 4: Check Non-Singularity for Each Combination
Now we need to check which combinations yield a non-zero determinant.

1. If \( a = \omega \) and \( c = \omega \):
\[ \text{det}(A) = 1 - \omega^2 - \omega + \omega^2 \cdot \omega^2 = 1 - \omega - \omega^2 + 1 = 2 - (\omega + \omega^2) = 2 - 1 = 1 \neq 0 \]

2. If \( a = \omega \) and \( c = \omega^2 \):
\[ \text{det}(A) = 1 - \omega^3 - \omega + \omega^2 \cdot \omega^2 = 1 - 1 - \omega + 1 = 1 - \omega \neq 0 \]

3. If \( a = \omega^2 \) and \( c = \omega \):
\[ \text{det}(A) = 1 - \omega^2 - \omega^2 + \omega^2 \cdot \omega = 1 - 2\omega^2 + \omega^2 = 1 - \omega^2 \neq 0 \]

4. If \( a = \omega^2 \) and \( c = \omega^2 \):
\[ \text{det}(A) = 1 - \omega^2 - \omega^2 + \omega^2 \cdot \omega^2 = 1 - 2\omega^2 + 1 = 2 - 2\omega^2 \neq 0 \]

### Conclusion
After checking all combinations, we find that all combinations yield a non-zero determinant. Thus, all 8 combinations are valid.

### Final Answer
The number of distinct non-singular matrices in the set \( S \) is:

**d. 8**