Various parameter-efficient fine-tuning (PEFT) methods, including LoRA and its variants, have gained popularity for reducing computational costs. However, there is often an accuracy gap between PEFT approaches and full fine-tuning (FT), and this discrepancy has not yet been systematically explored. In this work, we introduce a method for selecting sparse sub-matrices that aims to minimize the performance gap between PEFT vs. full fine-tuning (FT) while also reducing both fine-tuning computational costs and memory costs. We explored both gradient-based and activation-based parameter selection methods to identify the most significant sub-matrices for downstream tasks, updating only these blocks during fine-tuning. In our experiments, we demonstrated that SMT consistently surpasses other PEFTbaselines (e.g., LoRA and DoRA) in fine-tuning popular large language models such as LLaMA across a broad spectrum of tasks, while reducing the GPU memory footprint by 67% compared to FT. We also examine how the performance of LoRA and DoRA tends to plateau and decline as the number of trainable parameters increases, in contrast, our SMT method does not suffer from such issues.