[UPSC CSE 2016] Angular momentum, spin etc (Q293, 8+4+4+4=20M)

Write down the matrix representation of the three Pauli matrices \sigma_x, \sigma_y and \sigma_z. Prove that these matrices satisfy the following identities:

(i) [\sigma_x,\sigma_y]=2i\sigma_z

(ii) [\sigma^2\cdot\sigma_x]=0

(iii) (\vec{\sigma}\cdot\vec{A})(\vec{\sigma}\cdot\vec{B})=\vec{A}\cdot\vec{B}+i\vec{\sigma}\cdot(\vec{A}\times\vec{B}) if \vec{A} and \vec{B} commute with Pauli matrices.