Final answer:
Richard's inability to raise his arm despite understanding the command suggests that his primary motor cortex, which is responsible for initiating voluntary movements, has been affected by the stroke.
Step-by-step explanation:
The part of Richard's brain that has been affected by the stroke, given that he can understand the command to raise his arm but cannot originate the message to do so, is likely the primary motor cortex. The primary motor cortex is responsible for initiating voluntary motor movements. It is located in the precentral gyrus of the frontal lobe. Since the primary motor cortex's principle output is to stimulate spinal cord neurons to cause muscle contraction, damage here would prevent Richard from executing the movement to raise his arm even though he understands the command.