[UPSC IFoS 2010] Angular momentum, spin etc (Q322, 10M)

The electron spin operator \hat{s} can be expressed in matrix form in terms of the Pauli spin operator, \hat{\sigma} as \hat{\sigma} = 2\hat{s} where \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ +i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} Show that \sigma_x^2 = \sigma_y^2 = \sigma_z^2 = 1 \quad \text{and} \sigma_x \sigma_y = -\sigma_y \sigma_x = i \sigma_z \sigma_y \sigma_z = -\sigma_z \sigma_y = i \sigma_x \sigma_z \sigma_x = -\sigma_x \sigma_z = i \sigma_y