Homework 9, Math 152-205, additional problem:

(a) Let A,B be orthogonal.  Then AB is orthogonal if and only if
(AB)^T (AB) = I, where I is the identity matrix.  So we calculate

  (AB)^T (AB) = B^T A^T (AB) = B^T (A^T A) B = B^T I B = B^T B = I ,

and the condition is verified.

(b) The product of two reflections is a rotation.  This can be verified
by multiplying 

[  cos(a)    sin(a)  ]          [  cos(b)    sin(b)  ]
[                    ]   times  [                    ]
[  sin(a)   -cos(a)  ]          [  sin(b)   -cos(b)  ]

to get 

[ cos(a-b)   -sin(a-b) ]
[                      ]
[ sin(a-b)    cos(a-b) ]

(using  cos(a-b) = cos(a)cos(b)-sin(a)sin(b) and 
sin(a-b)=sin(a)cos(b)-sin(b)cos(a)  ).

Alternatively one can notice that every orthogonal 2 x 2 matrix has its
columns consisting of two unit length vectors that are orthogonal; so
if the first column is  [ cos(c)  sin(c) ]^T, the second column is
either  [ sin(c) -cos(c) ]^T  or  [ -sin(c) cos(c) ]^T.  This means that
a 2 x 2 matrix is orthogonal if and only if it is either a rotation or
a reflection.  Rotations are the ones with determinant 1, and reflections
are the ones with determinant -1; so the product of two reflections is an
orthogonal matrix with determinant 1, and is therefore a reflection.