[UPSC CSE 2021] Angular momentum, spin etc (Q267, 8+6+6=20M)

Using Pauli spin matrices prove that,

(i) \sigma_x\sigma_y+\sigma_y\sigma_x=0;\ \sigma_y\sigma_z+\sigma_z\sigma_y=0;\ \sigma_x\sigma_z+\sigma_z\sigma_x=0

(ii) \sigma_+\sigma_-=2(1+\sigma_z)

(iii) \sigma_\alpha+\sigma_\beta=i\sigma_\gamma where \alpha\ne\beta\ne\gamma