A linear algebra exercise

Anyone who has ever composed a written test on linear algebra knows that it often taken considerable effort to make the 'numbers come out right'. In fact, this problem may even be harder than the linear algebra problem itself. A particular subject where this problem arises, is eigenvalues of symmetric matrices. Usually we consider only matrices with integral coefficients. We know that 4 x 4-matrices tend to become too laborious. Since 2 x 2-matrices are not very exciting we have to confine ourselves to 3 x 3-matrices. We know that there are 3 real eigenvalues in this case. We cannot let the poor students solve irreducible cubic equations, so we might see to it that the eigenvalues are integers, which the students can recognize by inspection. Moreover, computing the eigenvalue equation of an arbitrary symmetric 3 x 3-matrix may be a little too stressful....