Infinite Geometric Series of Reciprocal Roots from Matrix Equation

To find the value of the infinite series, we first determine the relation between $a$, $b$, and $c$ using the given matrix equation:

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

This gives the 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):
$a + 8b + 7(-a - b) = 0$
$-6a + b = 0 \implies b = 6a$

Given that $b = 6$:
$6a = 6 \implies a = 1$
$c = -(1 + 6) = -7$

Thus, the quadratic equation $ax^2 + bx + c = 0$ becomes:
$x^2 + 6x - 7 = 0$

For the roots $\alpha$ and $\beta$, by Vieta's formulas:
$\alpha + \beta = -\frac{b}{a} = -6$
$\alpha \beta = \frac{c}{a} = -7$

Now, evaluate the term:
$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-6}{-7} = \frac{6}{7}$

Since $\left|\frac{6}{7}\right| < 1$, the infinite geometric series converges and its sum is:
$\sum_{n=0}^{\infty} \left(\frac{1}{\alpha} + \frac{1}{\beta}\right)^n = \sum_{n=0}^{\infty} \left(\frac{6}{7}\right)^n = \frac{1}{1 - \frac{6}{7}} = \frac{1}{\frac{1}{7}} = 7$

Hence, the correct option is (B).