 Solution

Each operator has trace one, so its characteristic polynomial is ${\lambda }^2-\lambda +\det \rho$. The first and last are diagonal; for the others, the symmetric and antisymmetric vectors diagonalize the matrix. Their eigenvalues areFor ${\rho }_1$, $(1,-1{)}^T/\sqrt{2}$ has eigenvalue one and $(1,1{)}^T/\sqrt{2}$ has eigenvalue zero. For ${\rho }_2$, those eigenvalues are $1/4$ and $3/4$, respectively. All four density operators are positive semidefinite, as these spectra confirm.