There are several questions about the backstory of Hamlet, one being the extent of Gertrude’s knowledge, or indeed any character’s actual knowledge, of the circumstances of elder Hamlet’s death. Gertrude is the most crucial for Hamlet’s dilemma, which I interpret as how he might avenge his father’s death without punishing his mother’s culpability in it. This is his external dilemma, one most concerned with the story and plot. There is a deeper dilemma, one of character that Hamlet faces. More on it later.
First to Gertrude. It is she that he damns first in his monologue following the revelation of the Ghost:
Yea, from the table of my memory
I’ll wipe away all trivial fond records,
All saws of books, all forms, all pressures past,
That youth and observation copied there;
And thy commandment all alone shall live
Within the book and volume of my brain,
Unmix’d with baser matter: yes, by heaven!
O most pernicious woman!
O villain, villain, smiling, damned villain!
“O villain” is a return to the calumny of Claudius, but Hamlet’s first expression of ire is telling. He will literally turn away from killing Claudius to go resolve the issue with his mother in Act 3. No matter his rationalization for his delay – it is only because he must deal with his mother’s guilt before he unleashes the swelling currents of vengeance. How else can he be sure to stay his hand in obedience to his dead father’s will?
Indeed, he almost kills her there in her bedroom, despite his pleading after the play for temperance. Polonius falls by his sword, and only the revelation of his identity shocks the joy of killing back into its place for Hamlet. It is a vital question for Hamlet, and for us the audience. Shakespeare understands that, and since his motive is the provocation of cathartic tragedy, he answers that question for us (but not for Hamlet).
Polonius comes to petition King Claudius with news: he has learned a possible motive for Hamlet’s recent madness, and since it propels Polonius to center stage, it is a fact as far as he is concerned. Polonius has told his daughter to stop seeing Hamlet, almost accusing her of being a stupid slut in a callous nobleman’s hands. Shakespeare has shown us the hypocrisy that gives Polonius cause for concern – it is behavior he expects and encourages from his son, and thus from any unmarried male member of nobility. Yet Hamlet has often shown his disdain of Polonius, and so Polonius wrapped in his narcissism believes Hamlet’s attentions to be about the two of them. When Ophelia describes a recent startling encounter between Hamlet and her, Polonius is stunned – Hamlet was actually in love with his daughter! And off he runs with his news.
And it is in the telling that all secrets come out. After Polonius has finished his rhetorical flourishes, there’s a slight pause in which Claudius digests this less-than-satisfying news:
Do you think ’tis this?
It may be, very likely.
Hath there been such a time–I’d fain know that–
That I have positively said ‘Tis so,’
When it proved otherwise?
Not that I know.
[Pointing to his head and shoulder]
Take this from this, if this be otherwise:
If circumstances lead me, I will find
Where truth is hid, though it were hid indeed
Within the centre.
Shakespeare has not confirmed Claudius’s guilt yet. Hamlet believes it with all his heart, and we should be extremely suspicious of the man. Claudius betrays himself here for us. “Do you think ‘tis this?” Claudius doesn’t wish to offend the old man, but he’s had far more opportunity to observe the mad Hamlet, and it never seems to be about spurned love whenever they are together. Yet Gertrude is surprisingly supportive (considering her treatment of Polonius earlier in the scene) and Polonius then springs his ace.
“Hath there been such a time – I’d fain know that –
That I have positively said ‘Tis so,’
When it proved otherwise?”
It’s far more dramatic to interpret this not as an old man puffing and preening, but as a crafty old man nailing Claudius to the ground with his knowledge of Claudius’s guilt. It is a dangerous game that both are playing here, and the stakes can only be one thing: Gertrude’s discovery of Claudius’s guilt.
How polite they are to each other here! And after the comedy, the audience will be drawn into this sudden snake fight. It is only our suspicion of Claudius that gives us the true import of the conversation. Gertrude’s innocence keeps her from seeing it in any other way but the preening old man.
So many inner world reveal themselves. Earlier, Gertrude ascribes her son’s madness to “his father’s death and our o’erhasty marriage.” This third blow to her son is seen through a loving mother’s eyes, and becomes the final link in her flattering unction. She will forever support Ophelia throughout the rest of the play, even admitting her wish for their marriage over her grave.
The circumstances of elder Hamlet’s death emerge clearly – Claudius acted alone, and Polonius discovered the truth (whether through inference or by his favorite method of fact gathering, concealment behind a curtain). Polonius confronted Claudius with this knowledge, and forever ensured his place in King Claudius’s court.
And Gertrude knows nothing untoward about the death of her husband. We as an audience learn this unreservedly at this point. It is knowledge we have that Hamlet does not, and because of it we can follow him down the path of vengeance while building up the emotions to be discharged in catharsis. We can watch him in his mother’s bedroom, drawing his sword yet again while he watches her observing her repentance, ready to send her to heaven – at once purging his desire to punish her and yet honoring his father’s wish. There only the reappearance of his father can stop him, and when Gertrude shows she is not haunted by his death (for those haunted by a death in Shakespeare will always see the ghost of their victim), Hamlet finally can accept that Gertrude had no knowing part in his father’s death and can relent. She has been tried, and she has passed her test. Only the circumstances unleashed by the death of Polonius can keep Hamlet from his vengeance now, and every step Hamlet takes from then on is one that brings him closer to a position in which he can take his vengeance and keep his kingdom from falling apart.