Doubting to have recited Surah Al-Fatihah

Q: While I was reciting the Surah (Qur’anic chapter) [after Surah Al-Fatihah] in the second Rak‘ah (unit of Prayer) of the Fajr (Dawn) Prayer on Monday 23/9/1411 A.H., I thought that I had not recited Surah Al-Fatihah. In fact, I am not sure when the thought came to me, it was either when I rose up for the second Rak‘ah, or when I started reciting the Surah or during recitation. So, I recited Al-Fatihah and followed it with some Ayat (Qur’anic verses) from the Surah that I had been reciting. I then performed Sujud-ul-Sahw (Prostration of Forgetfulness) before saying the Taslim (salutation of peace ending the Prayer). Upon concluding the Salah (Prayer), I reproached the Ma’mums (persons being led by an Imam in Prayer) - who were over fifty people - for not letting me know. After all people had left, two men stayed behind and told me that I had recited Surah Al-Fatihah twice, meaning that I was mistaken when I first thought that I had not recited it. What is the ruling on that Salah in which I recited Al-Fatihah twice (in the same Rak‘ah) and then performed Sujud-ul-Sahw?

A: If the situation is as you mentioned, then your Salah as well as the Salah of the Ma’mums is valid.May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.