Rotation Matrices
Part 4: Determinants and
Inverses of Rotation Matrices
- What is the determinant
of a 2-dimensional rotation matrix? Explain why all such matrices have
the same determinant.
- 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?
- 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.
- Explain the result of the
preceding step in terms of a formula for the inverse of a 2 x 2 matrix.
- 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.
- 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.
- Repeat the preceding step
with questions about columns instead of rows.
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