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...

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.