The matrix permanent belongs to the complexity class #P-Complete. It is gener- ally believed to be computationally infeasible for large problem sizes, and significant research has been done on approximation algorithms for the matrix permanent. We present an implementation and detailed runtime analysis of one such Markov Chain Monte Carlo (MCMC) based Fully Polynomial Randomized Approximation Scheme (FPRAS) for the matrix permanent which has previously only been described theo- retically and with big-Oh runtime analysis. We demonstrate that the constant factors hidden by the big-Oh analysis result in computational infeasibility. We explore the performance of the FPRAS implementation under relaxed sampling parameters to gauge the room for improvement in the probabilistic analysis of sampling parameter requirements for the FPRAS.