Matrix is a representation similar to vector in mathematics. If we have to investigate, the best explanation is that the matrix is a description of the mapping that represents the cartesian products of two finite sets and ordered sets as the domain.
Let me give you an example. All mappings from {1, 2} X{ 1, 2} to real number field r can be described by a second-order real square matrix.
{1, 2, ..., m} x {1, 2, ... n} to the real number field r can be described by an m×n matrix.
To describe in a detailed way. This is a more rigorous statement of the matrix itself.
But matrices are generally used as tools for algebraic representation in mathematics.
After defining various conventional operations for matrix in algebra, the matrix itself is related to the linear mapping between finite dimensional linear spaces. This is a very good tool. Moreover, in module theory, matrices over rings are also used to represent things similar to linear mappings, which is an important tool to help modules make elementary factorization theorems.
In other fields of algebra, matrices often appear as tools in various representations.
Moreover, we can define some structures on a set composed of a large class of matrices and give some examples of abstract structures.
The most classic are Lie groups and Lie algebras.
In fact, matrix is an important tool to study finite-dimensional algebra. Beginners can regard it as a generalized form of vector.