Evaluate Expression Involving Cube Roots of Unity from Matrix Equation

To find the value of the given expression, we first determine the relationship among $a$, $b$, and $c$ from the given matrix equation:

$$\left[\begin{array}{ccc} a & b & c \end{array}\right] \left[\begin{array}{ccc} 1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \end{array}\right] = \left[\begin{array}{ccc} 0 & 0 & 0 \end{array}\right]$$

Carrying out the matrix multiplication yields the following system of linear equations:

1. $a + 8b + 7c = 0$
2. $9a + 2b + 3c = 0$
3. $7a + 7b + 7c = 0 \implies a + b + c = 0$

From equation (3), we have:
$c = -(a + b)$

Substituting $c = -(a + b)$ into equation (1):
$a + 8b + 7(-a - b) = 0$
$-6a + b = 0 \implies b = 6a$

Then, substituting $b = 6a$ into the expression for $c$:
$c = -(a + 6a) = -7a$

Checking these values in equation (2):
$9a + 2(6a) + 3(-7a) = 9a + 12a - 21a = 0$
The relations are consistent.

Given that $a = 2$, we find:
$b = 6(2) = 12$
$c = -7(2) = -14$

We are given that $\omega$ is a non-real complex cube root of unity satisfying $x^3 - 1 = 0$ with $\text{Im}(\omega) > 0$. Hence:
$\omega^3 = 1 \quad \text{and} \quad 1 + \omega + \omega^2 = 0 \implies \omega + \omega^2 = -1$

Now, evaluate the powers of $\omega$:

• For $a = 2$:
  $\frac{1}{\omega^a} = \frac{1}{\omega^2} = \omega$

• For $b = 12$:
  $\frac{1}{\omega^b} = \frac{1}{\omega^{12}} = \frac{1}{(\omega^3)^4} = 1$

• For $c = -14$:
  $\frac{1}{\omega^c} = \omega^{-c} = \omega^{14} = (\omega^3)^4 \cdot \omega^2 = \omega^2$

Substitute these into the expression:
$\frac{3}{\omega^a} + \frac{1}{\omega^b} + \frac{3}{\omega^c} = 3\omega + 1 + 3\omega^2$
$= 1 + 3(\omega + \omega^2)$
$= 1 + 3(-1) = 1 - 3 = -2$

Thus, the correct option is A.