The matrices P and Q represent, respectively, clockwise rotations about (0, 0) through 45° and 90°, write down matrices P and Q, why P2=Q?

Here is the form I remember best. It is a general form for counterclockwise rotation by angle θ:

[cos(θ) -sin(θ)]

[sin(θ) cos(θ)]

A 45° clockwise rotation would be -45° counterclockwise, so let θ = -45°.

Do similarly for 90°.

The matrix P² would, of course, represent two successive clockwise rotations of 45°, so the net transformation would have to be a 90° clockwise rotation.