In a question I am given 4 vectors labelled $v_1$ to $v_4$ and a $3\times 3$ matrix $A$ with the following instructions:
Compute $Av_i$ for $i = 1, \dots , 4$ and determine which of the vectors $v_i$ are eigenvectors for the matrix $A$. Hence find the determinant and the characteristic polynomial of $A$.
How would I get a determinant using eigenvectors?
Assuming you got 3 different eigenvectors and the corresponding eigenvalues, the product of the eigenvalues is the determinant of $A$ since $\chi_A(X) = -\prod_{i=1}^3 (X- \lambda_i)=\det(A-XI_3)$ is the characteristic polynomial of $A$ (depending on your definition, you might have learnt $\chi_A(X) = \det(XI_3-A)$).