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1. Symmetry operations can be described by linear transformations r' = Ar, where r' = (x',y', z') and T' = (x, y, z), are spatial coordinates in R3, and A is a IR3 + 1R3 operator. Examine each the following symmetry operations, find the corresponding matrix representation of A and show that it is orthogonal: a) Inversion through the origin. b) Reflection in the yz-plane. c) n-fold rotation about the x-axis. (Note: an n-fold rotation is a rotation through an angle 0 = 27 ) d) n-fold improper rotation about the x-axis. (Note: an improper rotation about an axis is the combination of an n-fold rotation about the axis and a reflection in a plane perpendicular to that axis)