What Is the Date of Prophet Muhammad’s Birth and Death?
Question

What are the dates of the birth and death of the Messenger (blessings and peace of Allah be upon him)? I have seen many opinions concerning that; which is the correct opinion and what is the evidence in the light of the Quran and Sunnah?
Praise be to Allah.Different views on the day and month of Prophet Muhammad’s birth
The biographers and historians differed concerning the day and month of the birth of the Prophet (blessings and peace of Allah be upon him). This is something for which there is good reason, as it was not known what status this newborn would achieve; his situation was like that of any other newborn. Hence no one can state with certainty the date of his birth (blessings and peace of Allah be upon him). 
Dr. Muhammad at-Tayyib an-Najjar (may Allah have mercy on him) said: 
“Perhaps the reason for this difference of opinion is that when he was born, no one expected that he would attend such a status. For that reason, he did not live in the spotlight from the beginning of his life. When Allah decreed that the Messenger (blessings and peace of Allah be upon him) should begin to convey His message forty years after he was born, people began to seek out whatever memories they had about this Prophet, and they asked one another about every detail of his life. They were helped in that by what the Messenger (blessings and peace of Allah be upon him) himself narrated about incidents that he had experienced from his birth, as well as the reports narrated by his Companions and those who had also gone through those events with him. 
At that time the Muslims began to collect everything that they had heard about the life story of their Prophet (blessings and peace of Allah be upon him) in order to transmit it to the people down throughout the ages.” (Al-Qawl al-Mubeen fi Seerat Sayyid al-Mursaleen, p. 78)
Consensus on the year and day of Prophet Muhammad’s birth
Among the matters on which there is consensus with regard to his birth (blessings and peace of Allah be upon him) is the definition of the year and the day. 
1. With regard to the year:
It was the Year of the Elephant. Ibn al-Qayyim (may Allah have mercy on him) said: 
“There is no difference of opinion concerning the fact that he (blessings and peace of Allah be upon him) was born in Makkah, and that his birth was in the Year of the Elephant.” (Zad al-Ma'ad fi Hadiy Khayr al-‘Ibad, 1/76)
Muhammad ibn Yusuf as-Salihi (may Allah have mercy on him) said: 
“Ibn Ishaq (may Allah have mercy on him) said: (It was) the Year of the Elephant. 
Ibn Kathir said: This is the well-known view of the majority. 
Ibrahim ibn al-Mundhir al-Hizami, the Shaykh of al-Bukhari, said: This is the opinion concerning which none of the scholars has any doubt. Khalifah ibn Khayyat, Ibn al-Jazzar, Ibn Dihyah, Ibn al-Jawzi and Ibn al-Qayyim went further and narrated that there was consensus on this matter.” (Subul al-Huda wa’r-Rashad fi Seerat Khayr al-‘Ibad, 1/334, 335)
Dr. Akram Diya al-‘Umari (may Allah guide him) said: 
“In fact the reports that say otherwise all have flawed isnads; they suggest that he was born ten years, or twenty-three years, or forty years, after the Year of the Elephant. But the majority of scholars are of the view that he was born in the Year of the Elephant. This is supported by the modern study undertaken by both Muslim and Orientalist researchers who stated that the Year of the Elephant corresponds to the year 570 CE or 571 CE.” (As-Seerah an-Nabawiyyah as-Sahihah, 1/97) 
2. With regard to the day:
It was a Monday. He (blessings and peace of Allah be upon him) was born on a Monday, his mission began on a Monday and he died on a Monday. 
Abu Qatadah al-Ansari (may Allah have mercy on him) said: The Prophet (blessings and peace of Allah be upon him) was asked about fasting on Mondays and he said: “That is a day on which I was born and on it my mission began – or Revelation came to me.” (Narrated by Muslim, 1162)
Ibn Kathir (may Allah have mercy on him) said: 
Those who say that he was born on Friday 17th Rabee‘ al-Awwal were mistaken. 
“This was narrated by al-Hafiz Ibn Dihyah from what he read of the book I‘lam ar-Wara bi A‘lam al-Huda by one of the Shi‘ah. Then Ibn Dihyan stated why it was da‘if (weak), and it deserves to be classed as da‘if because it is contrary to the text.” (As-Seerah an-Nabawiyyah, 1/199)
Different views on defining the month and the day of the month of Prophet Muhammad’s birth
With regard to the point on which there is scholarly disagreement, that has to do with defining the month and the day of the month. We have come across many opinions concerning that, including the following: 
1. That he was born on 2nd Rabee ‘ al-Awwal.
Ibn Kathir (may Allah have mercy on him) said: 
“It was said that he was born on the second of the month. This was stated by Ibn ‘Abd al-Barr in al-Isti'ab, and was narrated by al-Waqidi from Abu Ma‘shar Nujayh ibn ‘Abd ar-Rahman al-Madani.” (As-Seerah an-Nabawiyyah, 1/199)
2. It was said that he was born on 8th Rabee‘ al-Awwal.
Ibn Kathir (may Allah have mercy on him) said: 
“It was said that he was born on the eighth of the month. This was narrated by al-Humaydi from Ibn Hazm, and was narrated by Malik, ‘Aqeel, Yunus ibn Yazeed and others from az-Zuhri from Muhammad ibn Jubayr ibn Mut‘im. Ibn ‘Abd al-Barr narrated that the historians regarded it as sahih; it was stated definitively by al-Hafiz al-Kabeer Muhammad ibn Musa al-Khawarizmi, and was regarded as most likely to be correct by al-Hafiz Abu’l-Khattan ibn Dihyah in his book at-Tanweer fi Mawlid al-Basheer an-Nadheer.” (As-Seerah an-Nabawiyyah, 1/199) 
3. It was said that he was born on 10th Rabee ‘ al-Awwal.
Ibn Kathir (may Allah have mercy on him) said: 
“It was said that he was born on the tenth of the month. This was narrated by Ibn Dihyah in his book, and was narrated by Ibn ‘Asakir from Abu Ja‘far al-Baqir. It was also narrated by Mujalid from ash-Sha‘bi.” (As-Seerah an-Nabawiyyah, 1/199)
4. It was said that he was born on 12th Rabee ‘ al-Awwal.
Ibn Kathir (may Allah have mercy on him) said: 
“It was said that he was born on the twelfth of the month. This was stated by Ibn Ishaq. It was narrated by Ibn Abi Shaybah in his Musannaf from ‘Affan from Sa‘eed ibn Mina that Jabir and Ibn ‘Abbas said: The Messenger of Allah (blessings and peace of Allah be upon him) was born in the Year of the Elephant on Monday 12 
th Rabee‘ al-Awwal; on Monday his mission began, on Monday he was taken up to heaven, on Monday he migrated, and on Monday he died.
This is the well-known opinion according to the majority. And Allah knows best.” (As-Seerah an-Nabawiyyah, 1/199) 
It was also said that he was born in Ramadan or in Safar or some other month. 
What appears to us to be the case is that the strongest opinion concerning the birth of the Prophet (blessings and peace of Allah be upon him) is that he was born between the eighth and the twelfth of Rabee‘ al-Awwal. Some Muslim mathematicians and astronomers have determined that the Monday in question was the ninth of Rabee‘ al-Awwal. It may be that this is another opinion, and it has some merit. That corresponds to 20th April 571 CE. This is what the authors of some modern books on Seerah (Prophet’s biography) regard as most likely to be correct, including Professor Muhammad al-Khudari and Safi ar-Rahman al-Mubarakfuri. 
Abu’l-Qasim as-Suhayli (may Allah have mercy on him) said: 
“The mathematicians say that his birth occurred in the solar month of April, on the twentieth of that month.” (Ar-Rawd al-Unuf, 1/282) 
Prof. Muhammad al-Khudari (may Allah have mercy on him) said: 
“The Egyptian astronomer Mahmoud Basha (d. 1885 CE) who was well-versed in astronomy, geography and mathematics, and wrote several books based on his research stated that it was on the morning of Monday 9th Rabee‘ al-Awwal, which corresponds to 20th April 571 CE. That corresponds to the first year following the elephant incident. He was born in the house of Abu Talib in Shi‘ab Banu Hashim.” (Nur al-Yaqeen fi Seerat Sayyid al-Mursaleen (p. 9). See also ar-Raheeq al-Makhtum, p. 41)
Date of the death of our Prophet Muhammad
With regard to the date of the death of our Prophet Muhammad (blessings and peace of Allah be upon him), there is no difference of opinion concerning the fact that it occurred on a Monday. The report narrated from Ibn Qutaybah, saying that it occurred on a Wednesday, is not correct. Perhaps what he meant is that the Prophet (blessings and peace of Allah be upon him) was buried on Wednesday, which is correct. 
With regard to the year of his death, there is no difference of opinion concerning the fact that it was in 11 AH. 
With regard to the month of his death, there is no difference of opinion concerning the fact that it was in Rabee‘ al-Awwal. 
Different views on the date of Prophet Muhammad’s death in Rabee` al-Awwal
With regard to the date of his death in that month, there is a difference of opinion among the scholars. 
1. The majority are of the view that it occurred on 12th Rabee‘ al-Awwal.
2. al-Khawarizmi was of the view that it occurred on 1st Rabee‘ al-Awwal.
3. Ibn al-Kalbi and Abu Makhnaf were of the view that it occurred on 2nd Rabee‘ al-Awwal. As-Suhayli was inclined to favour this view and al-Hafiz Ibn Hajar (may Allah have mercy on him) regarded it as most likely to be correct.
The most well-known view is that of the majority, which is that the death of the Prophet (blessings and peace of Allah be upon him) occurred on 12th Rabee‘ al-Awwal 11 AH.
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References
Ar-Rawd al-Unuf by as-Suhayli, 4:439-440.
As-Seerah an-Nabawiyyah by Ibn Kathir, 4:509.
Fath al-Bari by Ibn Hajar, 8:130. 
And Allah knows best.