GPU architectures have become popular for executing general-purpose programs. In particular, they are some of the most efficient architectures for machine learning applications which are among the most trendy and demanding applications nowadays.This paper presents SIMIL (SIMple Issue Logic for GPUs), an architectural modification to the issue stage that replaces scoreboards with a Dependence Matrix to track dependencies among instructions and avoid data hazards. We show that a Dependence Matrix is more effective in the presence of repetitive use of source operands, which is common in many applications. Besides, a Dependence Matrix with minor extensions can also support a simplistic out-of-order issue. Evaluations on an NVIDIA Tesla V100-like GPU show that SIMIL provides a speed-up of up to 2.39 in some machine learning programs and 1.31 on average for various benchmarks, while it reduces energy consumption by 12.81%, with only 1.5% area overhead. We also show that SIMIL outperforms a recently proposed approach for out-of-order issue that uses register renaming.