A density matrix is a mathematical representation used in quantum mechanics to describe the statistical state of a quantum system. It provides a way to account for both pure states, where a system is in a definite quantum state, and mixed states, where the system is in a combination of different states. The density matrix contains all the information needed to calculate the probabilities of various outcomes when measuring the system. In a density matrix, the diagonal elements represent the probabilities of the system being in specific states, while the off-diagonal elements indicate the coherence between those states. This framework is particularly useful in situations involving entangled systems, such as those described by quantum entanglement, and helps in understanding phenomena like quantum decoherence and quantum information theory.