The computation and memory-intensive nature of DNNs limits their use in many mobile and embedded contexts. Application-specific integrated circuit (ASIC) hardware accelerators employ matrix multiplication units (such as the systolic arrays) and dedicated nonlinear function units to speed up DNN computations. A close examination of these ASIC accelerators reveals that the designs are often specialized and lack versatility across different networks, especially when the networks have different types of computation. In this paper, we introduce a novel systolic array architecture, which is capable of executing nonlinear functions. By encompassing both inherent linear and newly enabled nonlinear functions within the systolic arrays, the proposed architecture facilitates versatile network inferences, substantially enhancing computational power and energy efficiency. Experimental results show that employing this systolic array enables seamless execution of entire DNNs, incurring only a negligible loss in the network inference accuracy. Furthermore, assessment and evaluation with FPGAs reveal that integrating nonlinear computation capacity into a systolic array does not introduce extra notable (less than 1.5%) block memory memories (BRAMs), look-up-tables (LUTs), or digital signal processors (DSPs) but a mere 13.3% - 24.1% more flip flops (FFs). In comparison to existing methodologies, executing the networks with the proposed systolic array, which enables the flexibility of different network models, yields up to 25.73x, 5.21x, and 1.54x computational efficiency when compared to general-purpose CPUs, GPUs, and SoCs respectively, while achieving comparable (83.4% - 135.8%) performance with the conventional accelerators which are designed for specific neural network models.