Rotation Matrices

Part 4: Determinants and Inverses of Rotation Matrices

1. What is the determinant of a 2-dimensional rotation matrix? Explain why all such matrices have the same determinant.
2. What can you say about determinants of the special 3-dimensional rotation matrices P, Q, R defined in Part 3? What about determinants of products of these matrices?
3. Compute the products AAT and ATA for the 2-dimensional rotation matrix A. What do you conclude about the transpose of a rotation matrix? Interpret this result in terms of the geometry of rotations.
4. Explain the result of the preceding step in terms of a formula for the inverse of a 2 x 2 matrix.
5. Does the conclusion of Step 3 hold for the special 3-dimensional rotation matrices P, Q, R? Explain your answer by using a formula for the inverse of any invertible matrix.
6. Each entry of a product of the form BBT is a dot product of a row of B with a column of BT -- but a column of BT is a row of B. Thus, the entries are dot products of rows of B with themselves and with each other. If B is any of the matrices A, P, Q, R, what is the dot product of a row of B with itself? What is the dot product of each row with a different row? Interpret your answers in terms of lengths of vectors and angles between vectors.
7. Repeat the preceding step with questions about columns instead of rows.

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