Does Gertrude think Hamlet is mad in act 3, scene 4 of Hamlet?

Gertrude's opinion on Hamlet's sanity varies throughout act 3, scene 4. She certainly believes him to be crazy as the scene begins, for she allows Polonius to spy on their conversation. After Hamlet accuses her of her sins, however, she probably recognizes his sanity. She doubts it again after Hamlet speaks with the ghost, and the audience is left unsure of her true opinion at the end of the scene.

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Gertrude believes—or at least very much fears—that Hamlet is mad through much of this scene. As the scene opens, she is jumpy around him, perceiving that he is in a very emotional state. When he says to her, speaking metaphorically, that he will hold up a "glass" or mirror where she "may see the inmost part of" herself, she takes his words literally. She fears he is going to cut her up or disembowel her in front of a mirror. We know this because she responds,
What wilt thou do? Thou wilt not murder me?
She then cries out in fear:
Help, help, ho!
This cry arouses Polonius to speak out from behind the arras in alarm. Hamlet, in a frenzy, stabs through the arras, killing Polonius.
It seems evident that Gertrude would not have taken Hamlet's words literally and cried out in fear if she was not deeply worried that he had completely lost his mind. Normally, a mother does not imagine her son would murder her and especially does not fear he would disembowel her in front of a mirror.
While she does accept that he speaks the truth when he accuses her of dirtying her soul in marrying Claudius, she again thinks Hamlet has gone mad when he starts speaking to his father's ghost. She sees nothing, and she cries out:
Alas, he’s mad!
She tells him the ghost is a figment of his imagination, saying,
This the very coinage of your brain.
Gertrude, at one and the same time, knows her son's words about her guilt in remarrying are true and yet fears he has gone over the edge into insanity.
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At the beginning of act 3, scene 4 of Hamlet, Queen Gertrude almost certainly believes that her son, Hamlet, has gone mad. His behavior has been so erratic and his speech so strange that anyone would think so. This is why she allows Polonius to spy on her meeting with her son. Gertrude is to be harsh with Hamlet and try to discover the reasons behind his madness.

As the scene continues, Hamlet makes no secret of what he thinks of Gertrude's actions. His near violence frightens Gertrude, perhaps convincing her further that her son has gone crazy. Polonius, hidden behind a curtain, cries for help, and Hamlet, thinking the spy is Claudius, stabs him. Gertrude is bewildered and horrified by what has happened.

Hamlet, however, does not let up on Gertrude. He makes her see her hypocrisy and her sin all the more clearly. What she has done, he implies, is real madness. Her passions have overtaken her. She has married her dead husband's brother. Gertrude does not want to hear more. Hamlet's words are “like daggers,” and they make her see the state of her soul. No insane person could speak in this way, and Gertrude likely no longer thinks her son to be mad.

Then the ghost of Hamlet's father appears, and Hamlet speaks to him. Gertrude, though, cannot see or hear the ghost, and once again, she questions her son's sanity as she witnesses Hamlet talking to empty air. Hamlet assures his mother that he is not crazy. His madness is only an act that he is using to exact his revenge. Gertrude promises not to tell Claudius. At this point, the audience does not know for sure whether Gertrude is convinced by Hamlet's words or not. He is certainly speaking sanely enough at this point, yet the image of him speaking to “nothing” must remain in Gertrude's mind. Still, she agrees to do as Hamlet says.

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After Hamlet mistakenly kills Polonius, he continues his rant against Gertrude. She keeps asking him to stop. Hamlet’s father, the Ghost, enters and speaks to Hamlet. Hamlet replies and Gertrude asks:

Alas, how is’t with you,

That you do bend your eye on vacancy

And with the incorporal air do hold discourse?

Again, Hamlet asks if she can see the Ghost. She cannot and the Ghost exits. At this point, she must think Hamlet is mad. But then Hamlet tells Gertrude that he is not mad (crazy). He is distraught by her and Claudius’ crime and this is why he speaks to the air (Ghost). He asks here to confess her sins and avoid getting into bed with Claudius. Hamlet then asks Gertrude to convince the king that he is mad. This is to prevent Claudius from supposing any plot of revenge.

I don’t think we can be sure about whether Gertrude thinks Hamlet is mad or not at this point. She saw him speak to the “incorporal air,” but he told her he was not mad. Later, Gertrude does what Hamlet asks and tells Claudius that Hamlet is mad. Since she goes along with this, we can assume that it is more likely that, by the end of this scene, Gertrude does not think Hamlet is mad.

Gertrude is a difficult character. It is never clear how much, if at all, she was involved with Claudius' plot in killing the king.

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