Matthew can cut $T$ into 16 equilateral triangles with side length 3. If he instead included a triangle of side 8, then let him include $a$ triangles of side length 3. He must include $12^{2}-8^{2}-3^{2} a=80-9 a$ triangles of side length 1. Thus $a \leq 8$, giving that he includes at least $$(80-9 a)+(a)+1=81-8 a \geq 17$$ total triangles, so 16 is minimal.