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Poor Hamlet. All he wants is a mom who bakes cookies for the PTA bake sale and stays true to his dad's memory—but instead, she marries her dead husband's brother, King Claudius. Is this really an act of betrayal to her husband's memory? Or is mom just not ready to be a widow shut up in some corner of the palace?
Oh, Gertrude. We've got a lot of questions about this lady. She's obviously a central figure in the play —Hamlet spends a whole lot of time dwelling on her incestuous marriage to Claudius —but we know practically nothing about her motivations or feelings. Was she having an affair with Claudius before the death of Old Hamlet? Does Gertrude know that Claudius killed her former husband? Why does she drink the poisoned wine her husband has prepared for her son? Does she know it's poisoned? Or, is she just really thirsty?
Let's take these one at a time.
First things first: Was Gertrude stepping out with Claudius while Old Hamlet was still alive? The Ghost all but accuses her of adultery and incest when he calls her new husband, Claudius, "that incestuous, that adulterate beast" (1.5.49). Okay, it seems clear Gertrude's guilty of adultery (cheating on one's spouse) right? Well, maybe not. In Shakespeare's day, "adulterate" could refer to any sexual sin (like incest), not just cheating.
But, if you really want to argue that Gertrude's a big old cheater, be sure to check out the ghost's emphasis on the marriage "vow" he made to Gertrude (1.5.56): he says that there was a "falling-off" from him to Claudius, which sure seems to imply that she was cheating on Old Hamlet while he was alive—and just maybe even plotting with Claudius.
Next question. Did Gertrude know her late husband was murdered by her new man? On the "No way!" side, the Ghost never accuses Gertrude of murder—just adultery. Also, Gertrude seems pretty surprised when Hamlet accuses her of "kill[ing] a king and marry[ing] with his brother" (3.4.35). (Although we might suggest that, if she doesn't know, she might be kind of dumb.)
On the other hand, we could also argue that Gertrude is surprised because, well, she's been caught. Take your pick.
Finally, does Gertrude know she's chugging poisoned wine in the play's final act?
Surprise! Again, Shakespeare doesn't exactly make things clear. She goes to drink; Claudius tells her not to, and she says: "I will, my lord; I pray you, pardon me" (5.2.318). Is she apologizing for drinking when he tells her not to? Does "I will" mean, "I'm going to take to control of my own destiny for once"? Or is she just thirsty?
In Laurence Olivier's film adaptation of Hamlet, Gertrude drinks knowingly, presumably to save her son from certain death. If she drinks on purpose, then she's the self-sacrificing mother Hamlet has always wanted her to be. But we're not convinced that the rest of the play has shown her to be self-sacrificing at all. If nothing else, this moment reminds us that Gertrude is much more complex than Hamlet understands; she's more than just morally "frail" (1.2.150).
Hamlet is unhealthily obsessed with his mother's sex life (which raises the question: is there any healthy way for a son to be obsessed with his mom's sex life?). Early on in the play, we learn that Gertrude's "o'erhasty" and incestuous marriage to Claudius has shaken up Hamlet's world, leaving him with a sense that the world is contaminated, like an "unweeded garden" that's "rank and gross in nature" (1.2.139; 140). In fact, he might even be more disturbed by Gertrude's sexuality than the news of his father's murder.
Even though the ghost warns Hamlet "Taint not thy mind, nor let thy soul contrive / Against thy mother aught. Leave her to heaven" (1.5.92-93), Hamlet just can't seem to leave his mother alone. One final question: does this tell us anything useful about Gertrude? Or does it tell us a whole lot more about Hamlet?