Exercise 2

In this exercise, we will consider $2\times 2$ matrices as defining matrix transformations.

1. 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 $A^{-1}$.

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

3. Find three more matrices $A$ that satisfy $A^2 = I$.