Large Language Models (LLMs) have demonstrated remarkable proficiency in a wide range of NLP tasks. However, when it comes to authorship verification (AV) tasks, which involve determining whether two given texts share the same authorship, even advanced models like ChatGPT exhibit notable limitations. This paper introduces a novel approach, termed InstructAV, for authorship verification. This approach utilizes LLMs in conjunction with a parameter-efficient fine-tuning (PEFT) method to simultaneously improve accuracy and explainability. The distinctiveness of InstructAV lies in its ability to align classification decisions with transparent and understandable explanations, representing a significant progression in the field of authorship verification. Through comprehensive experiments conducted across various datasets, InstructAV demonstrates its state-of-the-art performance on the AV task, offering high classification accuracy coupled with enhanced explanation reliability.