Oy The Pauli spin matrices used to describe particles with spin 1/2 (such as electrons) are 01 = = 1 (9) 2 = () Oz (a) Verify that y = y, y0z = 00y, (b) Verify that o=0 = 0 = 1, where I = (69) and oz is the identity matrix. (c) Denoting = I, show that tr(0,0;) I, show that tr(0,j) = 28ij for all i,j = {0, x, y, z}. Here, dij is a Kronecker delta function, and tr(A) = A is the trace of the matrix A (sum of all diagonal elements). (d) The three Pauli matrices r, Oy, z and the identity matrix = I form a linearly independent set that is a basis for all 22 matrices. Thus, any 22 matrix M (including complex matrices) can be written as a linear combination of those four matrices: M = Mu M2 M21 M22 = 1000+ A20x + Ayy + Azz. Derive the equations for the expansion coefficients A; for all i {0, x, y, z} in terms of matrix elements of M.