Raising hands in Salah in the positions mentioned in the question is a Sunnah of the Prophet (peace be upon him). It was authentically reported from
`Abdullah ibn `Umar (may Allah be pleased with him) that he said,
I saw that whenever the Messenger of Allah (peace be upon him) began Salah, he used to raise both his hands up to the shoulders. He used to do the same on pronouncing Takbir (saying: "Allahu Akbar [Allah is the Greatest]") for Ruku` and on raising his head from it, when he used to say, “Sami`a-l-lahu liman hamidah” (Allah listens to those who praise Him). However, he did not do so (i.e. raising his hands) in Sujud (prostration).
In another narration,
The Messenger of Allah (peace be upon him) (Part No. 6; Page No. 344) used to raise his hands up to the shoulders when beginning Salah."
Both Hadiths were narrated by
It was also authentically reported that
Ibn `Umar (may Allah be pleased with him) began Salah, he used to pronounce Takbir and raise his hands.
Ibn `Umar attributed this to the Prophet (peace be upon him).
This was also authentically reported in the Hadith narrated by
Abu Humayd Al-Sa`idy from the Prophet (peace be upon him).
As for placing the right hand on the left hand, it is one of the Sunan of Salah as well.
It was narrated by
Sahl ibn Sa`d (may Allah be pleased with him) that he said,
People were asked to place their right hands on their left hands in Salah.
Abu Hazim said, "As far as I know, this is attributed to the Messenger of Allah (peace be upon him)."
May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.