Calculate determinant from eigenvalue and eigenvectors

[Enemy of my enemy]

I have problem with this example:
We have matrix A with eigenvectors (1, 2, 1)^T (1, 1, 0)^T for eigenvalue λ = 7. And tr A = 2. We have to calculate determinant of A, but I dont know how I can do that. And can't find any help. I know how to calculate eigenvalue (or eigenvectors) from matrix, but opposite operation looks much harder.

 

[Romsek]

The determinant is the product of all the eigenvalues associated with each eigenvector.
(i.e. we included repeated eigenvalues in the product the number of times they are repeated)