Due to their superior results, Transformer-based models such as BERT have become de facto standards in many Natural Language Processing (NLP) applications. However, the intensive use of complex non-linear functions within the Transformer architecture impairs its computing efficiency and complicates corresponding accelerator designs, because non-linear functions are generally computation-intensive and require special hardware support. In light of this, we propose MA-BERT, which allows matrix arithmetic-only operations in Transformer-based NLP models and achieves efficient inference with negligible accuracy loss. Specifically, we propose four correlated techniques that include approximating softmax with a two-layer neural network, replacing GELU with ReLU, fusing normalization layers with adjacent linear layers, and leveraging knowledge transfer from baseline models. Through these techniques, we are able to eliminate the major non-linear functions in Transformer-based models and obtain MA-BERT with only matrix arithmetic and trivial ReLU operations without compromising on accuracy. With mainly regular matrix arithmetic operations, MA-BERT enables hardware-friendly processing on various computing engines, including CPUs and GPUs. Our experimental results show that MA-BERT achieves up to 27% and 41% reduction in inference time on CPU and GPU, respectively, with comparable accuracy on many downstream tasks compared to the baseline BERT models.