Matrix (mathematics)

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Matrix (mathematics)

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. Matrices have wide applications in engineering, physics, economics, and statistics as well as in various branches of mathematics. Historically, the use of matrices in Europe dates back to the 2nd century AD with the use by Chinese mathematicians, and by the 18th century, mathematicians like Gottfried Wilhelm Leibniz had begun to develop matrix theory to a greater extent. However, it was not until the 19th century that matrix theory was widely recognized and applied.

Definition and Notation

A matrix is typically denoted by a capital letter (A, B, C, ...) and its elements by a lowercase letter with two subscript indices (a_{ij}, b_{ij}, c_{ij}, ...), where i and j denote the row and column of the element, respectively. For example, a 2x3 matrix A would be written as:

\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix} \]

The size or dimension of a matrix is given by the number of rows and columns it contains, often denoted as m × n for a matrix with m rows and n columns.

Types of Matrices

Several special types of matrices are commonly used:

• A Square matrix is a matrix with the same number of rows and columns. An n × n matrix is known as a square matrix of order n.
• An Identity matrix, denoted by I, is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros.
• A Diagonal matrix is a square matrix in which the elements outside the main diagonal are all zero.
• A Zero matrix or null matrix is a matrix in which all the elements are zero.
• A Transpose of a matrix A, denoted by A^T, is a new matrix whose rows are the columns of A.

Matrix Operations

Matrices support several operations, including addition, subtraction, and multiplication. Notably, matrix multiplication is not commutative; that is, AB ≠ BA in general. The Determinant of a square matrix, denoted |A|, is a scalar value that can be computed from the elements of the matrix and has many applications in linear algebra, geometry, and differential equations.

Applications

Matrices are used in various fields for different purposes. In Linear algebra, they are used to solve systems of linear equations through techniques such as Gaussian elimination. In Computer graphics, matrices are used to perform transformations such as rotation, scaling, and translation on images. In Statistics, matrices are used in the analysis of multivariate data, and in Economics, they are used to model and solve economic problems involving multiple variables and equations.

See Also

• Linear algebra
• Determinant
• Eigenvalues and eigenvectors
• Matrix multiplication
• Inverse matrix