What is the ruling on the prayer of one who inadvertently added a third rak'ah to the Fajr prayer, then completed it and performed the prostration of forgetfulness while being doubtful about his action?

Persevering in adding to the prayer after being certain of its completion is impermissible and invalidates it. It is not rectified by the prostration of forgetfulness (sujud as-sahw). Therefore, you must return [to the correct number of units] upon remembering and perform the prostration of forgetfulness after the salutation (taslim). What you did was an error, and you must repent to Allah Almighty and repeat the prayer.