Efficient execution of SPARQL queries over large RDF datasets is a topic of considerable interest due to increased use of RDF to encode data. Most of this work has followed either relational or graph-based approaches. In this paper, we propose an alternative query engine, called gSmart, based on matrix algebra. This approach can potentially better exploit the computing power of high-performance heterogeneous architectures that we target. gSmart incorporates: (1) grouped incident edge-based SPARQL query evaluation, in which all unevaluated edges of a vertex are evaluated together using a series of matrix operations to fully utilize query constraints and narrow down the solution space; (2) a graph query planner that determines the order in which vertices in query graphs should be evaluated; (3) memory- and computation-efficient data structures including the light-weight sparse matrix (LSpM) storage for RDF data and the tree-based representation for evaluation results; (4) a multi-stage data partitioner to map the incident edge-based query evaluation into heterogeneous HPC architectures and develop multi-level parallelism; and (5) a parallel executor that uses the fine-grained processing scheme, pre-pruning technique, and tree-pruning technique to lower inter-node communication and enable high throughput. Evaluations of gSmart on a CPU+GPU HPC architecture show execution time speedups of up to 46920.00x compared to the existing SPARQL query engines on a single node machine. Additionally, gSmart on the Tianhe-1A supercomputer achieves a maximum speedup of 6.90x scaling from 2 to 16 CPU+GPU nodes.