How can the same algebraic methods apply to numbers, functions, and matrices?

In modern algebra, the same methods apply to numbers, functions, and matrices because they are all treated as instances of general algebraic structures. Rather than focusing on the specific nature of these objects, modern algebra emphasizes the study of their underlying properties and the relationships between them.1

This shift toward abstraction, pioneered in the early 20th century, allows mathematicians to use a unified set of tools—such as groups, rings, and fields—to analyze diverse mathematical entities. By defining a set of axioms (such as closure, associativity, and distributivity), mathematicians can determine how these systems behave regardless of whether the elements are individual numbers, complex polynomials, or matrices.12

For example, while matrix multiplication is not commutative—unlike the multiplication of ordinary numbers—matrices still satisfy other fundamental laws, such as the associative and distributive properties. By viewing these systems through the lens of abstract structures, mathematicians can apply consistent techniques to solve problems across different fields, effectively treating these varied objects as part of a single, coherent mathematical framework.314