4. Pauli's matrices. Suppose a 2 x 2 matrix X (not necessarily Hermitian, not unitary) is written as X = ao + 7.a, where the matrices o are Pauli's matrices and

ao, a1,2,3 are complex numbers in general. (a) Pauli's matrices satisfy the following anti-commutation relation adj + ajo₁ = 26j 12x2. Using the property, prove tr(oaj) = 26, and tr(o)= 0. (1 point) Do not use the explicit expression of Pauli's matrices. (b) How are ao and a (k = 1,2,3) related to X and o. (1 point) Do not use the explicit expression of Pauli's matrices. (c) Obtain ao and a in terms of the matrix elements Xij. (1 point) Use the explicit expression of Pauli's matrices.