Using numerical inaccurate numbers such as 0.3 are not a "proof" that something is wrong in the symbolic comp

Eigenvalues[\[Rho]Gpmatrix[\[Alpha], \[Beta], 3/10]] 

returns {0, 0, 7/20, 13/20} If you check more carefully by

Solve[CharacteristicPolynomial[\[Rho]Gpmatrix[\[Alpha], \[Beta], a], 
   x] == 0, x]

or

Solve[Det[\[Rho]Gpmatrix[\[Alpha], \[Beta], a] - 
    x IdentityMatrix[4]] == 0, x]

you will notice that the results only depend on the parameter a If you want a "numerical proof" just run

Table[Eigenvalues[\[Rho]Gpmatrix[\[Alpha], \[Beta], 0.3]], {\[Alpha], 
   0, 1, 0.1}, {\[Beta], 0, 1, 0.1}] // Chop

and see that the results do not depend on the values for alpha or beta yehuda