Commercial SRAM-based, field-programmable gate arrays (FPGAs) have the capability to provide space applications with the necessary performance, energy-efficiency, and adaptability to meet next-generation mission requirements. However, mitigating an FPGA's susceptibility to radiation-induced faults is challenging. Triple-modular redundancy (TMR) techniques are traditionally used to mitigate radiation effects, but TMR incurs substantial overheads such as increased area and power requirements. In order to reduce these overheads while still providing sufficient radiation mitigation, we propose the use of algorithm-based fault tolerance (ABFT). We investigate the effectiveness of hardware-based ABFT logic in COTS FPGAs by developing multiple ABFT-enabled matrix multiplication designs, carefully analyzing resource usage and reliability tradeoffs, and proposing design modifications for higher reliability. We perform fault-injection testing on a Xilinx Virtex-5 platform to validate these ABFT designs, measure design vulnerability, and compare ABFT effectiveness to other fault-tolerance methods. Our hybrid ABFT design reduces total design vulnerability by 99% while only incurring 25% overhead over a baseline, non-protected design.