Let a, b and c be three real numbers satisfying `[a" "b" "c"][{:(1,9,7),(8,2,7),(7,9,7):}]=[0"" "0""" "0]` and `alpha and beta` be the roots of the equation `ax^(2) + bx + c = 0`, then `sum_(n=0)^(oo) ((1)/(alpha)+(1)/(beta))^(n)`, is

Options

A. 6
B. 7
C. `(6)/(7)`
D. `oo`

Explanation

To solve the problem step by step, we start with the given matrix equation and the quadratic equation involving the roots α and β.

### Step 1: Solve the matrix equation
We have the matrix equation:

\[ [a \, b \, c] \begin{pmatrix} 1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 9 & 7 \end{pmatrix} = [0 \, 0 \, 0] \]

This implies that the rows of the matrix must lead to three equations:

1. \( a + 9b + 7c = 0 \)
2. \( 8a + 2b + 7c = 0 \)
3. \( 7a + 9b + 7c = 0 \)

### Step 2: Simplify the equations
We can simplify these equations to find the values of \( a \), \( b \), and \( c \).

From the first equation:
\[ a + 9b + 7c = 0 \quad \text{(1)} \]

From the second equation:
\[ 8a + 2b + 7c = 0 \quad \text{(2)} \]

From the third equation:
\[ 7a + 9b + 7c = 0 \quad \text{(3)} \]

### Step 3: Solve the equations
We can express \( c \) in terms of \( a \) and \( b \) using equation (1):
\[ c = -\frac{a + 9b}{7} \]

Substituting \( c \) into equations (2) and (3) will allow us to solve for \( a \) and \( b \).

### Step 4: Substitute \( c \) into equation (2)
Substituting \( c \) into equation (2):
\[ 8a + 2b + 7\left(-\frac{a + 9b}{7}\right) = 0 \]
This simplifies to:
\[ 8a + 2b - (a + 9b) = 0 \]
\[ 7a - 7b = 0 \implies a = b \]

### Step 5: Substitute \( a \) into equation (1)
Now substituting \( a = b \) into equation (1):
\[ a + 9a + 7c = 0 \implies 10a + 7c = 0 \implies c = -\frac{10a}{7} \]

### Step 6: Substitute \( a \) and \( c \) into equation (3)
Now substituting \( a = b \) and \( c = -\frac{10a}{7} \) into equation (3):
\[ 7a + 9a + 7\left(-\frac{10a}{7}\right) = 0 \]
This simplifies to:
\[ 16a - 10a = 0 \implies 6a = 0 \implies a = 0 \]
Since \( a = 0 \), we can conclude \( b = 0 \) and \( c = 0 \).

### Step 7: Find the roots of the quadratic equation
The quadratic equation is:
\[ ax^2 + bx + c = 0 \implies 0 = 0 \]
This indicates that the roots α and β are undefined.

### Step 8: Calculate the sum of the series
The expression we need to evaluate is:
\[ \sum_{n=0}^{\infty} \left( \frac{1}{\alpha} + \frac{1}{\beta} \right)^n \]
However, since α and β are undefined, we cannot compute this sum directly.

### Conclusion
The problem leads to a contradiction as the roots are undefined. Therefore, the sum cannot be computed.