###### Exercise2

This exercise concerns rotations and reflections in $$\real^2\text{.}$$

1. Suppose that $$A$$ is the matrix that performs a counterclockwise rotation in $$\real^2\text{.}$$ Draw a typical picture of the vectors that form the columns of $$A$$ and use the geometric definition of the determinant to determine $$\det A\text{.}$$

2. Suppose that $$B$$ is the matrix that performs a reflection in a line passing through the origin. Draw a typical picture of the columns of $$A$$ and use the geometric definition of the determinant to determine $$\det A\text{.}$$

3. As we saw in SectionÂ 2.6, the matrices have the form

\begin{equation*} A = \left[\begin{array}{rr} \cos \theta \amp -\sin\theta \\ \sin \theta \amp \cos \theta \\ \end{array}\right], \qquad B = \left[\begin{array}{rr} \cos(2\theta) \amp \sin(2\theta) \\ \sin(2\theta) \amp -\cos(2\theta) \\ \end{array}\right]\text{.} \end{equation*}

Compute the determinants of $$A$$ and $$B$$ and verify that they agree with what you found in the earlier parts of this exercise.

in-context