Is the prayer of an elderly woman who is sitting valid, given her inability to stand at the time of the opening takbir, and that performing the entire prayer while sitting is easier for her?

Standing in a fard (obligatory) prayer is a pillar for one who is able to do so. This includes standing for the opening takbir (takbirat al-ihram) and for the recitation of Al-Fatihah. Whoever finds apparent difficulty in standing, or fears an increase in illness, may pray sitting, and is not required to repeat the prayer. The slightest difficulty is not sufficient for this. So, if a woman finds it difficult to stand for the opening takbir, she may sit. However, if she is able to stand for it, she must do so, and then she may sit if standing during the recitation of Al-Fatihah becomes difficult for her.