Matrix

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is a fundamental concept in mathematics, particularly in linear algebra, and is used to represent and manipulate data, systems of equations, and transformations.

Matrices can be classified by their dimensions, denoted as m x n, where m represents the number of rows and n the number of columns. For example, a 3 x 2 matrix has three rows and two columns. Special types include square matrices (where m=n), which are significant in defining determinants, eigenvalues, and invertibility.

Operations on matrices include addition, subtraction, and multiplication. Matrix addition and subtraction are performed element-wise, provided the matrices are of the same dimension. Matrix multiplication involves taking the dot product of rows and columns, and it has specific rules and properties that distinguish it from scalar multiplication. Other key operations include finding the transpose, inverse (for square and invertible matrices), and calculating the determinant.

Matrices serve a broad range of applications across science and engineering. In computer graphics, they are used for transformations such as rotations and scaling. In systems engineering, matrices model complex systems of linear equations to find solutions. In data science, they underpin algorithms for machine learning and statistical analysis.

Matrices are also fundamental in theoretical mathematics, providing a framework for understanding transformations, vector spaces, and algebraic structures. Their properties and operations enable the concise expression of complex relationships and computations across diverse fields.