Earth system models (ESMs) have been rapidly developed in recent two decades to advance the research on climate change-carbon cycle feedback. However, the future land Carbon (C) and Nitrogen (N) storage change is still poorly projected by ESMs. It is imperative to better understand C and N dynamics in models. Toward this goal, we implemented the matrix approach to Community Land Model version 5 (CLM5), the land components of the Community Earth System Model version 2.0 (CESM2.0), to represent C and N transfer networks among both vegetation and soil pools in matrix equations. We verified that the matrix equations fully reproduce simulations of ecosystem C and N dynamics by the original model. The matrix approach calculates several critical diagnostics in C cycle, net primary productivity (NPP), net ecosystem productivity (NEP), residence time, chasing time, C storage capacity, and C storage potential. All these diagnostics carry ecological meanings, interpolate land C cycle response, and link to the observation. Results show that the chasing time, defined as the regression slope of NEP versus C storage potential, explains most of the spatial variation in land C uptake. Such variation in chasing time can be further traceable in the model parameters, biochemistry processes, or meteorological forcing. Constraints with observation on chasing time will further improve modeled carbon-climate response in future. The success to implement the matrix approach into CLM5, one of the most complicated land models in the world, proves the feasibility to employ the matrix approach on all other land surface models. Diagnostic variables calculated from this approach will greatly benefit future model inter-comparison projects. More comprehensive and consistent comparison analysis of land surface models will become possible.