For each of the following geometric operations in the plane, find a \(2\times 2\) matrix that defines the matrix transformation performing the operation.

  1. Rotates vectors by \(180^\circ\text{.}\)

  2. Reflects vectors in the vertical axis.

  3. Reflects vectors in the line \(y=-x\text{.}\)

  4. Rotates vectors counterclockwise by \(60^\circ\text{.}\)

  5. First rotates vectors counterclockwise by \(60^\circ\) and then reflects in the line \(y=x\text{.}\)