[UPSC CSE 2009] Angular momentum, spin etc (Q325, 20M)

Show that the Pauli Spin Matrices obey the following relations : (i) \text{Tr} (\sigma_x) = \text{Tr} (\sigma_y) (ii) \det (\sigma_y) = \det (\sigma_z) (iii) The eigenvalues of \sigma_z and \sigma_x are the same. (iv) Write down the y-component of the spin angular momentum matrix corresponding to an antineutrino.