The presence of multiple time scales in chemical reaction systems introduces stiffness, necessitating the use of implicit methods for solving the ordinary differential equation (ODE) system. For large stiff systems solved by implicit methods, solution of the linear system can dominate the computation time. It has been observed that ODE systems modeling chemically reacting processes often yield sparse linear systems. In this work we propose model reduction techniques by reducing the linear system matrices and improving their sparsity. Using a sparse linear solver (KLU) on the reduced linear system, we investigate the effect of our matrix reductions on the simulation efficiency. We implement two schemes for matrix reduction: 1) Threshold-based reduction; 2) Michaelis-Menten reduction. The threshold-based reductions delete matrix elements based on their absolute value, while Michaelis-Menten reduction modifies the matrix based on underlying chemical properties. We examine the use of the reduced matrix both directly in the direct linear solver, and as a preconditioner in an iterative linear solver. Results for different reduction schemes are provided and compared against the dense unreduced implementation.