If B = [[5, 2α, 1], [0, 2, 1], [α, 3, -1]] is the inverse of a 3x3 matrix A, then the sum of all values of α for which det(A) + 1 = 0, is?

Correct option is C.
|B| = 5(-5 - 3) - 2α(0 - α) + 1(0 - 2α) = -40 + 2α² - 2α
Since B is the inverse of A, |B| = 1/|A|
Therefore, |A| = 1/|B| = 1/(-40 + 2α² - 2α)
Given that det(A) + 1 = 0, we have |A| = -1
So, 1/(-40 + 2α² - 2α) = -1
This implies -40 + 2α² - 2α = -1
2α² - 2α - 39 = 0
Let's find the roots using the quadratic formula:
α = (-b ± √(b² - 4ac)) / 2a
α = (2 ± √(4 - 4(2)(-39))) / 4
α = (2 ± √(4 + 312)) / 4
α = (2 ± √316) / 4
α = (2 ± 2√79) / 4
α = (1 ± √79) / 2
The sum of the roots is (1 + √79)/2 + (1 - √79)/2 = 1
Therefore, the sum of all values of α is 1.