Two$-$spin system

Consider a system with two spins vec$(S{)}_1$ and vec$(S{)}_2,$ both with spin $\frac{1}{2}.$ Solve the eigenvalues and the eigenstates

for the Hamiltonian

hat$(H)=$hat$(vec(S){)}_1*hat(vec(S){)}_2$

Hint: Use the basis $|S,S_z,S_1,S_2$:$.$

For each of the eigenstates, what are its total $S$ and total $S_z$ quantum numbers?

The eigenstates can be classified to one singlet state and three triplet states. Which one is singlet and which

three are triplets?

Please answer this question!

Proof that all the eigenstates of hat$(S{)}^2$ are also eigenstates of the total hat$(S{)}_z$ operator, where hat$(S{)}_z=$hat$(S{)}_{1z}+$hat$(S{)}_{2z}.$