General matrix multiplication (GEMM) is a ubiquitous computing kernel/algorithm for data processing in diverse applications, including artificial intelligence (AI) and deep learning (DL). Recent shift towards edge computing has inspired GEMM architectures based on unary computing, which are predominantly stochastic and rate-coded systems. This paper proposes a novel GEMM architecture based on temporal-coding, called tuGEMM, that performs exact computation. We introduce two variants of tuGEMM, serial and parallel, with distinct area/power-latency trade-offs. Post-synthesis Power-Performance-Area (PPA) in 45 nm CMOS are reported for 2-bit, 4-bit, and 8-bit computations. The designs illustrate significant advantages in area-power efficiency over state-of-the-art stochastic unary systems especially at low precisions, e.g. incurring just 0.03 mm^2 and 9 mW for 4 bits, and 0.01 mm^2 and 4 mW for 2 bits. This makes tuGEMM ideal for power constrained mobile and edge devices performing always-on real-time sensory processing.