College Algebra Exam Review 209

College Algebra Exam Review 209 - trices with determinant 1...

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4.1. ROTATIONS OF REGULAR POLYHEDRA 219 (c) Compute the matrix of rotation through 2±=3 about the vector 2 4 1 1 1 3 5 ; and explain why the answer has such a remarkably simple form. 4.1.2. Show that the midpoints of the edges of the tetrahedron are at the six points ˙ O e 1 ; ˙ O e 2 ; ˙ O e 3 : Show that the matrix of the rotation by ± about any of the 2–fold axes of the tetrahedron is a diagonal matrix with two entries equal to ± 1 and one entry equal to 1 . Show that the set of these matrices generates a group of order 4. 4.1.3. Show that the matrices computed in Exercises 4.1.1 and 4.1.2 gen- erate the group of rotation matrices of the tetrahedron; the remaining ma- trices can be computed by matrix multiplication. Show that the rotation matrices for the tetrahedron are the matrices 2 4 ˙ 1 0 0 0 ˙ 1 0 0 0 ˙ 1 3 5 ; 2 4 0 0 ˙ 1 ˙ 1 0 0 0 ˙ 1 0 3 5 ; 2 4 0 ˙ 1 0 0 0 ˙ 1 ˙ 1 0 0 3 5 ; where the product of the entries is 1. That is, there are no ± 1 ’s or else two ± 1 ’s. 4.1.4. Show that the group of rotational matrices of the tetrahedron is the semidirect product of the group V consisting of diagonal permutation ma-
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Unformatted text preview: trices with determinant 1 (which is a group of order 4) and the cyclic group of order 3 generated by the permutation matrix R D 2 4 0 0 1 1 0 0 0 1 0 3 5 . 4.1.5. (a) If T denotes the group of rotation matrices for the tetrahedron and S is any rotation matrix for the cube that is not contained in T , show that the group of rotation matrices for the cube is T [ T s . (b) Show that the group of rotation matrices for the cube consists of signed permutation matrices with determinant 1, that is, matrices with entries in f 0; 1 g with exactly one nonzero entry in each row and in each column, and with determinant 1. (c) Show that the group of rotation matrices for the cube is the semidi-rect product of the group V consisting of diagonal permutation matrices with determinant 1 (which is a group of order 4) and the group of 3by3 permutation matrices....
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