The matrix operators corresponding to rotations Rr (0) and Ry (0) through an angle @ about the r and y axes are given by 0 0 cos 0 0 sing RT (0) = 0 cos 8 - sin 0 and Ry (0) = 0 1 0 O sin 0 cos 0 - sin 0 0 cose (a) Show that the matrix corresponding to a rotation through 0 about the r-axis, followed by a rotation through 02 about the y-axis, is given by cos 62 sin 61 sin 02 sin 02 cos #1 R(01, 02) = 0 cos #1 - sin 02 - sin 62 sin 0, cos 02 cose, cos 02 Do R. (01) and Ry (62) commute? (b) Write an expression for the inverse matrix R-(01, 02) in terms of R. (0) and R,(0) and hence confirm explicitly the relation R-1 = RT, which holds for any orthogonal matrix and show that det [R(0, )R(02 )] = 1 in this case.Which of the matrices below are: (i) symmetric, (ii) orthogonal, (iii) unitary or (iv) Hermitian? Use the matrix that has none of these properties to construct (v) an anti-symmetric matrix and (vi) an anti-Hermitian matrix. 2 1 + 27 1- 27 A = B = 1 - 2i 0 1 + 2i 3 1 1- i 2i C = -3i -21 1 D = 1 + 3i 2 3 V2 0 cos 0 - sin 8 0 E = sind cos O 0 0