Matrix multiplication is a core computational part of deep learning and scientific workloads. The emergence of Matrix Cores in high-end AMD GPUs, a building block of Exascale computers, opens new opportunities for optimizing the performance and power efficiency of compute-intensive applications. This work provides a timely, comprehensive characterization of the novel Matrix Cores in AMD GPUs. We develop low-level micro-benchmarks for leveraging Matrix Cores at different levels of parallelism, achieving up to 350, 88, and 69 TFLOPS for mixed, float, and double precision on one GPU. Using results obtained from the micro-benchmarks, we provide a performance model of Matrix Cores that can guide application developers in performance tuning. We also provide the first quantitative study and modeling of the power efficiency of Matrix Cores at different floating-point data types. Finally, we evaluate the high- level programmability of Matrix Cores through the rocBLAS library in a wide range of matrix sizes from 16 to 64K. Our results indicate that application developers can transparently leverage Matrix Cores to deliver more than 92% peak computing throughput by properly selecting data types and interfaces.