Large Language Models (LLMs) offer the potential to automate hiring by matching job descriptions with candidate resumes, streamlining recruitment processes, and reducing operational costs. However, biases inherent in these models may lead to unfair hiring practices, reinforcing societal prejudices and undermining workplace diversity. This study examines the performance and fairness of LLMs in job-resume matching tasks within the English language and U.S. context. It evaluates how factors such as gender, race, and educational background influence model decisions, providing critical insights into the fairness and reliability of LLMs in HR applications. Our findings indicate that while recent models have reduced biases related to explicit attributes like gender and race, implicit biases concerning educational background remain significant. These results highlight the need for ongoing evaluation and the development of advanced bias mitigation strategies to ensure equitable hiring practices when using LLMs in industry settings.