In an absorbing Markov chain, the fundamental matrix provides critical insights into the system's behavior before absorption occurs. It is defined as N=(I−Q)−1N = (I - Q)^{-1}, where QQ represents the transition matrix for the transient states. The fundamental matrix captures the expected number of times the system will be in each transient state, starting from a given transient state, before being absorbed. This information is essential for understanding the dynamics of the process, such as determining the expected number of steps to absorption and the probabilities of visiting specific transient states along the way. Consequently, the fundamental matrix is a powerful tool for analyzing long-term behavior and decision-making in stochastic systems.