Alzheimer's dementia (AD) is a neurodegenerative disorder with cognitive decline that commonly impacts language ability. This work extends the paired perplexity approach to detecting AD by using a recent large language model (LLM), the instruction-following version of Mistral-7B. We improve accuracy by an average of 3.33% over the best current paired perplexity method and by 6.35% over the top-ranked method from the ADReSS 2020 challenge benchmark. Our further analysis demonstrates that the proposed approach can effectively detect AD with a clear and interpretable decision boundary in contrast to other methods that suffer from opaque decision-making processes. Finally, by prompting the fine-tuned LLMs and comparing the model-generated responses to human responses, we illustrate that the LLMs have learned the special language patterns of AD speakers, which opens up possibilities for novel methods of model interpretation and data augmentation.