Exercise 2
This exercise concerns rotations and reflections in \(\real^2\text{.}\)
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 \(\det A\text{.}\)
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 \(\det A\text{.}\)

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.