My mother is obliged to perform Tayammum (dry ablution) so that she can read the Quran from the Mushaf (Quran book). After having performed Tayammum, she sometimes orders me to do things, so does this speaking invalidate her dry ablution?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

Speaking does not invalidate Tayammum (dry ablution), rather it is invalidated by the nullifiers of ablution, or by being able to perform ablution with water after not being able to do so, or by the presence of water after one was lacking it. 

Allaah Knows best.