Step-by-Step Solution:

1. Understand the Problem We are given a matrix template where a, b, and c can be either ω or ω², where ω is a non-real cube root of unity. The set S contains all such matrices that are non-singular. A matrix is non-singular if its determinant is non-zero. We need to find the number of distinct matrices in S.

2. Properties of Cube Roots of Unity We know that for a non-real cube root of unity ω:

   • ω³ = 1
   • 1 + ω + ω² = 0

3. Define the Matrix and Calculate its Determinant Let the given matrix be A: $A = \left[ {\begin{matrix} 1 & a & b \\ \omega & 1 & c \\ {{\omega ^2}} & \omega & 1 \\ \end{matrix} } \right]$ The determinant of A, det(A), is calculated by expanding along the first row: $\det(A) = 1(1 \cdot 1 - c \cdot \omega) - a(\omega \cdot 1 - c \cdot \omega^2) + b(\omega \cdot \omega - 1 \cdot \omega^2)$ $\det(A) = (1 - c\omega) - a(\omega - c\omega^2) + b(\omega^2 - \omega^2)$ $\det(A) = 1 - c\omega - a\omega + ac\omega^2 + b(0)$ $\det(A) = 1 - \omega(a+c) + ac\omega^2$ Notice that the determinant does not depend on the value of b.

4. Condition for Non-Singular Matrix For the matrix to be non-singular, we must have det(A) ≠ 0. $1 - \omega(a+c) + ac\omega^2 \neq 0$

5. Analyze Possible Cases for a and c The variables a and c can each take values from the set {ω, ω²}. This gives us four possible pairs for (a, c):

   • Case 1: a = ω, c = ω $\det(A) = 1 - \omega(\omega + \omega) + (\omega)(\omega)\omega^2$ $\det(A) = 1 - 2\omega^2 + \omega^4$ Since ω³ = 1, we have ω⁴ = ω. So, $\det(A) = 1 - 2\omega^2 + \omega$ Using the property 1 + ω = -ω²: $\det(A) = -\omega^2 - 2\omega^2 = -3\omega^2$ Since ω ≠ 0, we have det(A) = -3ω² ≠ 0. So, the matrix is non-singular in this case.

   • Case 2: a = ω, c = ω² $\det(A) = 1 - \omega(\omega + \omega^2) + (\omega)(\omega^2)\omega^2$ Using the property ω + ω² = -1 and ω³ = 1: $\det(A) = 1 - \omega(-1) + (\omega^3)\omega^2$ $\det(A) = 1 + \omega + 1 \cdot \omega^2 = 1 + \omega + \omega^2$ $\det(A) = 0$ So, the matrix is singular in this case.

   • Case 3: a = ω², c = ω This is symmetric to Case 2 with respect to a and c. The determinant expression 1 - ω(a+c) + acω² is also symmetric in a and c. $\det(A) = 1 - \omega(\omega^2 + \omega) + (\omega^2)(\omega)\omega^2$ $\det(A) = 1 - \omega(-1) + (\omega^3)\omega^2 = 1 + \omega + \omega^2 = 0$ So, the matrix is singular in this case.

   • Case 4: a = ω², c = ω² $\det(A) = 1 - \omega(\omega^2 + \omega^2) + (\omega^2)(\omega^2)\omega^2$ $\det(A) = 1 - 2\omega^3 + \omega^6$ Since ω³ = 1, ω⁶ = (ω³)² = 1: $\det(A) = 1 - 2(1) + 1 = 0$ So, the matrix is singular in this case.

6. Count the Number of Non-Singular Matrices The matrix is non-singular only when a = ω and c = ω. The value of the determinant is independent of b. The variable b can be either ω or ω². This gives us two distinct non-singular matrices:

   1. For (a, b, c) = (ω, ω, ω)
   2. For (a, b, c) = (ω, ω², ω)

   Therefore, the number of distinct matrices in the set S is 2.

7. Conclusion The total number of possible matrices is 2 × 2 × 2 = 8. We found that 6 of them are singular and 2 are non-singular. The set S consists of these 2 non-singular matrices. The correct option is A.