Is it permissible to shake hands with non-mahram?

The listener (Dr. M. M.) from Sudan sent a message containing a set of questions:
In his first question, he asks and says: What is the ruling in the Qur'an and the Sunnah regarding a man who shakes hands with female relatives and places a piece of cloth on his hand to separate his hand from the women's hands, or a woman who likewise places a barrier on her hand when shaking hands? Is there anything in the hadith of the Prophet (peace and blessings be upon him) regarding this? We ask for clarification?
Answer: It is obligatory for the believing man and woman to leave that, for shaking hands is not permissible with non-mahram. As for shaking hands with his sister, or his aunt, or his maternal aunt, or his wife, there is nothing wrong with it. As for shaking hands with his cousins, or his maternal cousins, or his maternal aunt's daughters, or his neighbors, or the like of that, this is not permissible, even if he places a rag, or she places a rag; because of the saying of the Prophet (peace and blessings be upon him): When he took their pledge of allegiance and a woman extended her hand to him, he said: And he is the model to be followed (peace and blessings be upon him). Allah said : [Al-Ahzab: 21]. And Aisha (may Allah be pleased with her) said: -meaning: in the pledge of allegiance- (peace and blessings be upon him). And because shaking hands with women is a means to fitnah, just as looking at them is a means to fitnah, so shaking hands is a means to fitnah; therefore, Allah has forbidden it to block the door of fitnah. Yes.