A woman went to the mosque to pray in a congregation. She followed the Imam just before he bowed. She said the opening invocation (a certain supplication said at the beginning of the prayer before reciting Al-Fatihah), and then, the Imam bowed. She followed him and bowed but did not recite Al-Fatihah. Shall she repeat the first Rak’ah as she did not recite Al-Fatihah, depending on the prophetic hadith which says: “He who does not recite Fatihat Al-Kitab (Al-Fatihah) is not credited with having observed prayer.” or not?

Answer

Praise Be to Allah and Peace Be Upon His Messenger.
Referring to your question, reciting Al-Fatihah in each Rak’ah of a prayer is a pillar of prayer. This is the opinion of the majority of Muslim scholars. However, if a latecomer follows the Imam while he (the Imam) bows, they can skip reciting it only to follow the Imam. The majority of scholars agreed that a Rak’ah is not counted when the follower misses bowing with the Imam, and thus, they must make it up. In contrast, Zufar, who followed Abu Hanifa, gave a different opinion. He said that if the latecomer follows the Imam while the Imam is raising up his head after bowing, the Rak’ah is counted. This is what Zufar copied from An-Nawawi’s The Explanation of Al-Muhathab.  
Undoubtedly, the correct opinion is the opinion of the majority of scholars.
How can a follower recognize that the Rak’ah is counted?
The condition is that the follower must at least bow (while the Imam is bowing) and put his palms on his knees. Other scholars added that the follower must feel a little relaxed as he is bowing. This relaxation, according to Hanbali Scholars, equals the time needed to say one praise. 
Therefore, if the Imam raises his head after bowing and you had not followed him, you must make up that Rak’ah after the Imam finishes. Moreover, the follower cannot bow while the Imam is standing, but they must follow him and do what he does. 
If a Muslim comes late, and the Imam is standing before bowing, the follower must consider the time. If he (the follower) can say the opening invocation and recite Al-Fatihah, he can say both of them. If he recognizes that the time allows him just to say one of them, he must recite Al-Fatihah only. If there is little time in which he can recite Al-Fatihah, but he wastes it saying the opening invocation, and then the Imam bows, this Rakah is not counted. 
Thus, if the latecomer follows the Imam while he is standing before bowing, he is not allowed to say the opening invocation unless he is certain that he can say it and recite Al-Fatihah. Otherwise, he must recite Al-Fatihah only. This is because saying the opening invocation is desired while reciting Al-Fatihah is a must. 
If the latecomer follows the Imam while the Imam is bowing or prostrating, he must not say the opening invocation as these situations have their own praises, and the opening invocation also has its own place too. It must be said in the first Rak’ah after the Opening Takbir and before reciting Al-Fatihah. 
Imam An-Nawawi says in his book, Al-Majmoo’, if the latecomer follows the Imam while the Imam is not standing before bowing, he should not say the opening invocation. If he follows the Imam recognizing that he can say the opening invocation, the appeal for refuge (Isti’athah) and recite Al-Fatihah, he can say it; otherwise, he must recite Al-Fatihah.
Finally, you should make up the Rak’ah; otherwise, you should make up the whole prayer.