In Game of Thrones Season 2 Episode 7, this exchange happens after Tywin finds his guardsman murdered with a dart poisoned with Wolfsbane:
Tywin Lannister: I don't like mutton.
Arya Stark: ... I- I'll bring something else-
TL: Leave it. (walks over to the table) You hungry?
AS: (shakes her head quickly) No.
TL: Of course you are. (gestures to the dish of bread and mutton) Eat.
AS: I'll... eat in the kitchen, later.
TL: (impatiently) It's bad manners to refuse a Lord's offer. (walks around her, offers her a knife) Sit- eat. (Arya sits, takes the knife, and- after hesitating starts eating ravenously) You're small, for your age- I suppose you've been underfed your whole life.
I could believe that he genuinely dislike mutton and likes Arya enough to have her eat before him (instead of bringing him more food), while he goes hungry.
However, isn't it very much possible that he suspected this highly educated girl of unknown noble parentage, dressed as a boy, to have something to do with the assassination? Only someone intelligent would have used a sophisticated poison, and the use of a dart could indicate that the assassin isn't a good fighter. He then insists that she eat before him to test her reaction should she have poisoned his food as well.
Is this covered in the books/elsewhere?