[UPSC CSE 2019] Angular momentum, spin etc (Q278, 15M)

Define Pauli spin matrices \sigma_x, \sigma_y and \sigma_z. Using these definitions, prove the following:

(i) \sigma_x^2=\sigma_y^2=\sigma_z^2=1

(ii) \sigma_x\sigma_y=i\sigma_z;\ \sigma_z\sigma_x=i\sigma_y;\ \sigma_y\sigma_z=i\sigma_x