We propose Row-Column Fine-Tuning(RoCoFT), a parameter-efficient fine-tuning method for large language models based on updating only a few rows and columns of the weight matrices in transformers. Through extensive experiments with medium-sized LMs like RoBERTa and DeBERTa, and larger LMs like Bloom-7B, Llama2-7B, and Llama2-13B, we show that our method gives comparable or better accuracies than state-of-the-art Parameter-Efficient Finetuning methods while also being more memory and computation-efficient. We also study the reason behind the effectiveness of our method with tools from neural tangent kernel theory. We empirically demonstrate that our kernel, constructed using a restricted set of row and column parameters, is numerically close to the full-parameter kernel and gives comparable classification performance. Ablation studies are conducted to investigate the impact of different algorithmic choices, including the robustness of RoCoFT to any selection of rows and columns, as well as the optimal rank for the effective implementation of our method.