What is the ruling on shaking hands with a non-mahram woman without desire, and shaking hands with an elderly non-mahram woman who has lost her desire?

It is not permissible for a man to shake hands with a non-mahram (definition) woman, such as his paternal cousin. Some scholars have permitted shaking hands with an old woman who is not desired, due to the absence of fear of temptation. However, the Malikis and Shafi'is prohibited touching a non-mahram woman without differentiating between young and old. The preponderant opinion is to prohibit shaking her hand, especially in this era where morals have deteriorated.