What could be a possible motivation for Gertrude's actions after the death of her husband? Does she know about her husband's murder? And why marry Claudius?
During the big confrontation between Hamlet and his mother, it becomes apparent that she knew nothing about the murder of her husband by Claudius. (See Act 3, Scene 4) If Claudius proposed to her, she would be in a delicate position. If she refused, she would be like a poor relation in the castle for the remainder of her life, totally dependent on Claudius's good will. Also, Claudius would have no reason to promise that her son Hamlet would be the heir to his throne. So Hamlet would also be in the position of a poor relation. Claudius would want to marry somebody. He needed a queen to strengthen his own position. He might make a diplomatic marriage with the daughter of a foreign ruler. In such a case, he would undoubtedly marry a young woman, and she would most likely have children who would take precedence over Hamlet in line for the crown. Claudius might even marry Ophelia, who would consent if her father ordered her to marry him--and Polonius, of course, would be delighted to have his daughter become the queen of Denmark. A new queen would make Gertrude's and Hamlet's positions even worse. Gertrude might have felt she had no choice but to say yes without even hesitating.
gertrude was possibly not involved in king hamlet's murder because the ghost told hamlet not to harm his mother , this dialogue clearly shows that she was not involved in the murder. however she might have married cladius because of lust which is denoted by hamlet several times.