Large Language Models (LLMs) have demonstrated exceptional performances in a wide range of natural language processing tasks. However, their success does not always extend to machine translation, particularly in challenging scenarios such as translating low-resource languages. This study investigates the multilingual capability of LLMs, with a case study on Irish, an extremely low-resource language, focusing on translation tasks between English and Irish. We propose a dynamic, efficient language adaptation framework for English-centric LLMs, which involves layer-specific adjustments and subsequent fine-tuning for machine translation. Our findings highlight several key insights: (1) different layers in the LLM serve distinct functions such as language understanding and task reasoning, (2) effective translation requires extensive pre-training on both source and target languages, and (3) targeted fine-tuning for machine translation leads to significant improvements of 36.7% for English to Irish and 133.4% for Irish to English compared to the previous state-of-the-art.