A 2x2 matrix has eigenvalues $\lambda_1 = 3$ and $\lambda_2 = 5$. What is $\det(A)$?

B. 15

$\det(A) = \lambda_1\lambda_2 = 3 \times 5 = 15$. The determinant is the PRODUCT of the eigenvalues and the trace is their sum, so 8 is the trace and 2 is their difference, neither is the determinant. The product rule is why a zero eigenvalue forces a zero determinant: one factor kills it, which is singularity seen through the eigenvalues.