The matrix permanent belongs to the complexity class #P-Complete. It is generally 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 theoretically and with big-Oh runtime analysis. We demonstrate by analysis and experiment that the constant factors hidden by previous big-Oh analyses result in computational infeasibility.