For a $3 \times 3$ matrix $A$ with eigenvalues $\lambda_1, \lambda_2, \lambda_3$, suppose $\text{tr}(A) = \lambda_1 + \lambda_2 + \lambda_3 = 12$ and $\text{tr}(A^2) = \lambda_1^2 + \lambda_2^2 + \lambda_3^2 = 50$. What is the value of $2\sum_{i < j} \lambda_i \lambda_j$?