Matrix groups are mathematical structures that consist of sets of matrices that can be combined through operations like addition and multiplication. These groups follow specific rules, such as having an identity element and inverses for each member, which allows for consistent manipulation of the matrices. In the context of linear algebra, matrix groups are often used to study transformations and symmetries in various spaces. A common example is the general linear group, which includes all invertible matrices, showcasing how these groups can represent complex systems in fields like physics and computer science.