Despite the patriarchal society, the eighteenth-century novel suggests strongly that women have choice and agency and "being as an independent agent." Examine this claim based on a comparative analysis of Austen's novels Pride and Prejudice and Persuasion. Please answer with a detailed analysis of BOTH Pride and Prejudice AND Persuasion.

A good place to start would be to compare the two main characters of the novels in question: Pride and Prejudice's Elizabeth Bennet and Persuasion's Anne Eliot. Despite living in a patriarchal system, both women exhibit agency and independence.
Elizabeth Bennet shows agency and independence in refusing to marry Mr. Collins, even though this is an advantageous match for her economically. As Mr. Collins so tactlessly points out, he is willing to do her the favor of taking her on, despite her lack of a dowry. Further, since Mr. Collins is heir to Longbourn, the estate where Elizabeth and her sisters have grown up, Elizabeth's marriage to him would keep the home in the family and ensure that the Bennet mother and sisters have a roof over their heads after their father dies. Nevertheless, Elizabeth steadfastly refuses him: she despises Mr. Collins and nothing would induce her to become his wife. She shows her strength in standing up to her mother, who exposes her daughter to ceaseless verbal abuse because she is so upset at Elizabeth's decision. Elizabeth displays independence both in defying her mother and by not letting herself be manipulated into an unhappy marriage by a patriarchal system that privileges male inheritance.
Elizabeth also defies patriarchy when she refuses Mr. Darcy's first marriage proposal. To his shock, as he fully expects her to fall in line with social norms and be honored that he, the important Lord Darcy, has proposed to the penniless Elizabeth Bennet, the arrogance of his offer makes her blow up in rage, as does his admission that he helped to separate Bingley and Jane. Elizabeth shows her power and independence as she expresses her fury toward a titled man (Lord Darcy) worth 10,000 pounds a year. He couldn't be more surprised, as he is used to having, like Miss Bingley, women fawn over him.
Likewise, Anne Eliot shows agency and independence in choosing her marriage partner. Although she initially capitulates to Lady Russell's persuasive advice that she not marry Wentworth because his future economic situation is uncertain, Anne later comes to regret that decision and determines to think for herself in the future. We learn, for instance, that she turned down a marriage proposal from her sister Mary's husband Charles Musgrove.
Like Elizabeth Bennet, Anne has no dowry. She also receives very little attention or esteem from her father or older sister. She is seen as a worthless throwaway to them. A marriage to someone like Charles would have been quite advantageous for Anne, keeping her close to home and raising her status from aging spinster to wife of a gentleman. This is especially true as Charles is a reasonable person and not a complete buffoon like Mr. Collins. Nevertheless, Anne refuses to marry a man she does not love.
Even when the price is high, both Elizabeth and Anne show they will follow their hearts and hold on to the right to make decisions for themselves. They will not, as some other women do, compromise their values in order to elevate themselves in the patriarchal order. Love with both women must come first.