This paper proposed and compared two FPGA (field programmable gate array)-based solutions to the inversion of sparse matrices. The solutions leveraged the sparseness of extremely large matrices in real-time electromagnetic transient (EMT) simulation. The FPGA implementation of sparse LU decomposition and modified Sherman-Morrison-Woodbury(SMW) formula in matrix inversion is presented. This resource-aware implementation tackled the resource bottleneck on FPGA-based sparse matrix inversion. Through a novel one matrix structure storage scheme, we finally achieved resource-reduced on the premise of maintaining speed and accuracy. Although the solutions are tailored for power electronics simulation systems, they can also be universalized in many other sparse-matrix-led fields due to its versatility. The resource consumption, performance, and accuracy are evaluated and compared. Our solution is proven to perform excellently.