Let a, b and c be three real numbers satisfying the matrix equation:

[[a, b, c], [1, 9, 7], [8, 2, 7]] * [[1], [8], [7]] = [[0], [0], [0]]

Let b = 6, with a and c satisfying the above matrix equation. If α and β are the roots of the quadratic equation ax² + bx + c = 0, then Σn=0∞ (α + β)n is:

Since a, b and c satisfy the matrix equation:

[[a, b, c], [1, 9, 7], [8, 2, 7]] * [[1], [8], [7]] = [[0], [0], [0]]

So, we get the equations:

a + 8b + 7c = 0
9a + 2b + 3c = 0
7a + 7b + 7c = 0 or a + b + c = 0

Since b = 6, so the equations become:

a + 48 + 7c = 0 (1)
9a + 12 + 3c = 0 (2)
a + 6 + c = 0 (3)

Subtracting (3) from (1), we get 42 + 6c = 0, which simplifies to c = -7

Using equation (3), we get a + 6 - 7 = 0, so a = 1.

So, we have a = 1, b = 6, c = -7

The quadratic equation is x² + 6x - 7 = 0. The sum of the roots α and β is given by -b/a = -6/1 = -6.

The sum Σn=0∞ (α + β)n is a geometric series with first term 1 and common ratio (α + β) = -6. Since the absolute value of the common ratio is greater than 1, the series diverges and the sum is ∞.