Is it permissible for a Muslim man to speak with a woman in the following situations: inquiring about the family's well-being, medical purposes, buying and selling, getting acquainted for marriage, and inviting her to Islam? And what is the evidence for this?

The permissibility of speaking with a non-mahram woman is conditional upon it being for a need, and that the speech be serious, concise, and devoid of flirtatiousness or softness in tone. It must also be from behind a screen, based on the Almighty’s saying: (And when you ask [his wives] for something, ask them from behind a screen. That is purer for your hearts and their hearts), and His saying: (So do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate words). The words must be appropriate, not rejected by the Sharia or by sound dispositions, and free from any suspicion.

The matters that necessitate speaking with a non-mahram woman are not limited to a specific number; rather, they arise whenever there is a need, while adhering to the aforementioned Sharia guidelines.