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.