A matrix commutator is a mathematical operation used in linear algebra, defined for two matrices, A and B. It is expressed as [A, B] = AB - BA, where AB is the product of the two matrices in that order, and BA is the product in the reverse order. The commutator measures how much the two matrices fail to commute, meaning how the order of multiplication affects the result. Matrix commutators are particularly important in quantum mechanics, where they help describe the relationships between observable quantities. For example, the commutator of the position operator X and momentum operator P is fundamental in understanding the uncertainty principle, highlighting the intrinsic limitations in simultaneously measuring certain pairs of physical properties.