### 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
**AA**^{T}
and **A**^{T}A 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
**BB**^{T} is a dot product of a row of **B** with a column
of **B**^{T} -- but a column of **B**^{T} 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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