Commuting Matrices

One of the first fundamental facts students learn in linear algebra is that matrix multiplication is generally not commutative: for square matrices A and B, we do not usually have AB=BA. This naturally leads to the question: under what conditions do matrices commute? While the identity matrix and the zero matrix commute with every matrix, there are many other matrices that share this property under certain structural constraints. In this talk, we investigate when two matrices commute and how these conditions relate to the structure of the matrices themselves. Using Jordan Normal Form and Jordan Decomposition, we characterize classes of matrices whose entries and algebraic structure guarantee that AB=BA.