If $\alpha, \beta \neq 0$, and $f(n) = \alpha^n + \beta^n$ and $ \begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) &1+f(4) \end{vmatrix} = K(1 - \alpha)^2 (1 - \beta)^2 (\alpha - \beta)^2$, then $K$ is equal to

Question

( A ) $\alpha \beta$
( B ) $-1$
( C ) $1$
( D ) $\frac{1}{\alpha \beta}$