Narrated by al-Bukhaari from the hadeeth of ‘Abdullaah ibn ‘Abbaas (may Allaah be pleased with him), that the people used to recite dhikr out loud when they finished the obligatory prayers at the time of the Prophet (peace and blessings of Allaah be upon him). He said: I used to know when they finished (the prayer) by that, when I heard it. 

In another Narration: I never use to know the ending of the prophet’s prayer except by Takbeer.

Ash-Shaykh al Uthaymeen, may Allaah have mercy upon him, said:

“As for what some people understood (from this hadeeth) that after giving salaam one should say ”Allaah Akbar” then this (understanding) is far ( from being correct) due to the hadeeth of the Sahaabee Thawbaan and the Hadeeth of A’ishah which both shows that the prophet, alaihi as-salaatu was salaam, would begin (his dhikr after the five daily prayers) with istigfaar, seeking forgiveness and saying “Allaahummaa antaa as-Salaam wa minka as-Salaam” after giving Tasleem ( exiting from the prayer)…”

Reference: Sharhul Bukaari (404/3), Print: al-Maktabul Islaamiyah.
Ash-Shaykh Ahmad an-Najmee, may Allaah have mercy upon him, said: 

“It was said ‘that which could be extracted from this Hadeeth is that it is recommended to make Takbeer after giving Salaam (from the prayer) but however this is Da’eef (a weak understanding). What is possible is that what was intended by the word ”takbeer” is that which is said when saying Tasbeeh and Tahmeed (saying Subanallaah, Allaahu akbar and alhamduliah 33 times each).”

Reference: Sharu Umdatil Ahkaam, (304/2 ), Print: Daarul Minhaaj.

Ibn Hajar, may Allaah have mercy upon him, said “…It is as if they use to begin (their dhikr) with Takbeer( 33 times) after the prayer before saying Tasbeeh and Tahmeed( 33 times each).”

Also this is the opinion of ash-Shaykh Albaani and many other Scholars of our time.

Ibn Battaal, may Allaah have mercy upon him, said: ” it (saying takbeer three times after giving salaams from the prayer) has not been narrated on any of the Salaf…” He, may Allaah have mercy upon him, also mentioned that the opinion of Imam Maalik is this act is an act of innovation.

Reference: Fat-hul Baaree , (421-420/2), Print: Daarus Salaam.

Also ash- Shaykh Ahmad An-Najmee inclines to the opinion that this act is an act of innovation.

Reference: Sharu Umdatil Ahkaam, (304/2), Print: Daarul Minhaaj.

Also Ibn Katheer, may Allaah have mercy upon him, has mentioned the first to perform this act was Ma’muun one of the rulers of the past; he would say” allahu akbar” three times after the obligator prayers with a loud voice and would depart upon that.

Reference: al-Bidaayah wan-Nihaayah.

Conclusion: Is that we Muslims should not perform any act of worship except with a proof from the Quraan or Sunnah , so we should not say immediately after the prayer “ Allaahu Akbar” loudly three times due to it not having a proof from the Quraan and Sunnah .

Here is Shaykh Albani’s speech pertaining to saying Takbeer loudly after the prayer:

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~ Abu Fajr AbdulFattaah bin Uthman