Are the prayers in which Al-Fatihah was recited only in the first two rak'ahs, and a surah in the third and fourth rak'ahs, and the final Tashahhud was started with "Allahumma salli 'ala Muhammad" instead of "At-tahiyyatulillah," invalid and must be repeated?

Among the pillars of prayer are the recitation of Al-Fatihah in every rak'ah and the final tashahhud, according to the majority of scholars. Omitting a pillar, even out of ignorance, invalidates the prayer and necessitates its reperformance. There is another opinion, chosen by Ibn Taymiyyah, which states that reperformance is not obligatory. The first opinion is more cautious and absolves one's responsibility. Therefore, we advise you to investigate the prayers you performed without Al-Fatihah in every rak'ah or without the final tashahhud and to re-perform them. If you are unable to do so, then act based on what you most likely believe to be true. You may also adopt the view that Al-Fatihah is obligatory but not a pillar (the view of Abu Hanifah), or that the pillar is the sitting for the tashahhud itself, not the tashahhud recitation (the view of the Malikis), as some jurists have permitted adopting a weaker opinion after the act has occurred and its rectification is difficult. Leaving an obligation out of ignorance does not invalidate the prayer.