If you apply the matrix to the vector $(1,0,0)$ you get $(1/2, -\sqrt{3}/2,0)$ which means that the matrix rotates vectors $60 ^{\circ}$ clockwise about the $z$ axis. We are told that both $(a_x', a_y', a_x')$ and $(a_x, a_y, a_z)$ represent the same vector. Therefore, the reference frame $S$ must be rotated $60 ^{\circ}$ counterclockwise about the $z$ axis so that $(1/2, -\sqrt{3}/2,0)$ represents the same vector as $(1,0,0)$ in the initial reference frame. Therefore, answer (E) is correct.

Solution to 2001 Problem 75