The marriage between Gertrude and Claudius is one of convenience and expediency. Neither one loves the other, but both realize it is in their interests to marry each other. Marrying his late brother's widow and the queen helps legitimize Claudius's claim to the throne. After all, if Gertrude accepts his right to be king, who is going to challenge him? Marrying Gertrude also prevents her from supporting her son, Hamlet, as king, something Claudius would like to avoid.
For Gertrude, marriage to Claudius allows her to remain queen of Denmark. It allies her with a powerful man and gives her a replacement for her beloved husband.
However, a marriage based on the partners using each other for their own gains is not likely to last, and the marriage is strained, which is shown when Gertrude confesses to Hamlet that Claudius is not the man her late husband was and when Claudius, at the end, is content to let Gertrude drink the chalice of poisoned wine.
The marriage is destructive as well, because it...
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