Is my prayer valid, even though I doubted my recitation of Al-Fatiha in the first or second rak'ah and completed the prayer and performed the prostration of forgetfulness (sujud al-sahw)?

The more authoritative opinion is that the Fatiha is a pillar of the prayer, and the prayer is not valid without it. If the worshipper (ma'mum) doubts having recited it before bowing (rukuk), he must recite it, because its place is still available. However, if the doubt occurs after bowing, he must follow the Imam and perform an additional rak'ah after the Imam's salutation. He is not required to perform sujud sahv (prostration of forgetfulness). If the worshipper re-recites the Fatiha after doubting it and before bowing, his prayer is valid. But if the doubt occurs after bowing and he does not perform an additional rak'ah after the Imam's salutation, his prayer is invalid.