How to other characters treat or respond to Hamlet?

Hamlet requests that his mother (Queen Gertrude) lie and tell Claudius that he is in fact mad. What we don't really know here is if her response is to do as he's asked (does she believe he is not mad?), or is her response that of a mother who believes her son has gone off the deep end. We never really find out......

"Mad as the sea and wind, when both contend

Which is the mightier: in his lawless fit,

Behind the arras hearing something stir,

Whips out his rapier, cries, 'A rat, a rat!'

And, in this brainish apprehension, kills

The unseen good old man."

Act IV / Scene I