Determinants multiply nicely, so
This proves II.
To see that III may be false, use the definition of an eigenvalue. If
is an eigenvalue of A, then its corresponding eigenvector
is an eigenvalue of A^2. This gives us a hint: If the distinct eigenvalues of A are opposites, we might only get one eigenvalue for A^2. Let's try constructing a 2x2 matrix with eigenvalues 1 and -1. We'll just put those values on the diagonal of a triangular matrix:
Then A^2=I, which has only 1 as an eigenvalue.