The algorithmic and implementation principles are explored in gainfully exploiting GPU accelerators in conjunction with multicore processors on high-end systems with large numbers of compute nodes, and evaluated in an implementation of a scalable block tridiagonal solver. The accelerator of each compute node is exploited in combination with multicore processors of that node in performing block-level linear algebra operations in the overall, distributed solver algorithm. Optimizations incorporated include: (1) an efficient memory mapping and synchronization interface to minimize data movement, (2) multi-process sharing of the accelerator within a node to obtain balanced load with multicore processors, and (3) an automatic memory management system to efficiently utilize accelerator memory when sub-matrices spill over the limits of device memory. Results are reported from our novel implementation that uses MAGMA and CUBLAS accelerator software systems simultaneously with ACML  for multithreaded execution on processors. Overall, using 940 nVidia Tesla X2090 accelerators and 15,040 cores, the best heterogeneous execution delivers a 10.9-fold reduction in run time relative to an already efficient parallel multicore-only baseline implementation that is highly optimized with intra-node and inter-node concurrency and computation-communication overlap. Detailed quantitative results are presented to explain all critical runtime components contributing to hybrid performance.