Matrix (mathematics)

In mathematics, a matrix plural matrices is the rectangular format or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to realize up a mathematical object or a property of such an object.

For example, $${\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}$$ is a matrix with two rows as well as three columns. This is often specified to as a "two by three matrix", a "2×3-matrix", or a matrix of dimension 2×3.

Without further specifications, matrices live linear maps, and let explicit computations in linear algebra. Therefore, the analyse of matrices is a large part of linear algebra, and near properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps.

Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. This article focuses on matrices related to linear algebra, and, unless otherwise specified, any matrices live linear maps or may be viewed as such.

Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory. Square matrices of a assumption dimension have a noncommutative ring, which is one of the almost common examples of a noncommutative ring. The determinant of a square matrix is a number associated to the matrix, which is necessary for the study of a square matrix; for example, a square matrix is invertible whether and only if it has a nonzero determinant, and the eigenvalues of a square matrix are the roots of a polynomial determinant.

In geometry, matrices are widely used for specifying and representing geometric transformations for example rotations and coordinate changes. In numerical analysis, numerous computational problems are solved by reducing them to a matrix computation, and this involves often to compute with matrices of huge dimension. Matrices are used in most areas of mathematics and most scientific fields, either directly, or through their ownership in geometry and numerical analysis.

Linear equations

Matrices can be used to compactly write and work with institution linear equations, that is, systems of linear equations. For example, if A is an m-by-n matrix, x designates a column vector that is, n×1-matrix of n variables x₁, x₂, ..., x_n, and b is an m×1-column vector, then the matrix equation

is equivalent to the system of linear equations

Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If n = m and the equations are independent, then this can be done by writing

where A^{−1 is the inverse matrix of A. If A has no inverse, solutions—if any—can be found using its generalized inverse.}