###### Exercise 2

In this exercise, we will consider \(2\times 2\) matrices as defining linear transformations.

Write the matrix \(A\) that performs a \(45^\circ\) rotation. What geometric operation undoes this rotation? Find the matrix that perform this operation and verify that it is \(A^{-1}\text{.}\)

Write the matrix \(A\) that performs a \(180^\circ\) rotation. Verify that \(A^2 = I\) so that \(A^{-1} = A\text{,}\) and explain geomtrically why this is the case.

Find three more matrices \(A\) that satisfy \(A^2 = I\text{.}\)