What is the idea of Islam about Mut'ah? Has The prophet Mohammad done it or not?
According to the Sunni view, Mut’ah marriage was allowed in early Islam, but later on the rules were changed. Similar to how alcohol was also allowed in early Islam, then later the rules for alcohol were also changed.
Have a read of this hadith regarding the rule change for mut'ah marriage:
It was narrated from al-Rabee’ ibn Sabrah al-Juhani that his father told him that he was with the Messenger of Allaah (peace and blessings of Allaah be upon him) who said, “O people, I used to allow you to engage in mut’ah marriages, but now Allaah has forbidden that until the Day of Resurrection, so whoever has any wives in a mut’ah marriage, he should let her go and do not take anything of the (money) you have given them.”
Narrated by Muslim, 1406.
Source of fatwa from which I found that hadith: islamqa.com