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Matrix In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. Matrices are described by the field of matrix theory. They can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra. The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (written m × n) and m and n are called its dimensions. The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is commonly said that an m-by-n matrix has an order of m × n ("order" meaning size). Two matrices of the same order whose corresponding entries are equivalent are considered equal. The entry that lies in the i-th row and the j-th column of a matrix is typically referred to as the i,j, or (i,j), or (i,j)-th, or (i,j)th entry of the matrix. Again, the row is always noted first, then the column. Almost always upper-case letters denote matrices, while the corresponding lower-case letters, with two subscript indices, represent the entries. For example, the (i,j)th entry of a matrix A is most commonly written as ai,j. Alternative notations for that entry are A[i,j] or Ai,j. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (nonitalic), to further distinguish matrices from other variables. Following this convention, A is a matrix, distinguished from A, a scalar. An alternate convention is to annotate matrices with their dimensions in small type underneath the symbol, for example, for an m-by-n matrix. We often write or to define an m × n matrix A. In this case, the entries ai,j are defined separately for all integers 1 ≤ i ≤ m and 1 ≤ j ≤ n. In some programming languages, the numbering of rows and columns starts at zero. Texts which use any such language extensively frequently follow that convention, so we have 0 ≤ i ≤ m-1 and 0 ≤ j ≤ n-1. A matrix where one of the dimensions equals one is often called a vector, and interpreted as an element of real coordinate space. An m × 1 matrix (one column and m rows) is called a column vector and a 1 × n matrix (one row and n columns) is called a row vector. An matrix is a function where is any non-empty set. is the Cartesian product of sets We say that matrix is a matrix over the set and . Important thing to note is that, if we want to have matrix algebra, the set must be a ring and matrix must be a square matrix. Since the set of all square matrices over a ring is also a ring, matrix algebra is usually called matrix ring. Since this article mainly considers matrices over real numbers, matrices shown here are actually functions The matrix or is a matrix. The element a2,3 or is 7. In terms of the mathematical definition given above, this matrix is a function and, for example, and The matrix is a matrix, or 9-element row vector. Two or more matrices of identical dimensions m and n can be added. Given m-by-n matrices A and B, their sum A+B is the m-by-n matrix computed by adding corresponding elements: For example: Another, much less often used notion of matrix addition is the direct sum. Scalar multiplication Given a matrix A and a number c, the scalar multiplication cA is computed by multiplying every element of A by the scalar c (i.e. ). For example: Matrix addition and scalar multiplication turn the set with real entries into a real vector space of dimension of all m-by-n matrices . Matrix multiplication Multiplication of two matrices is well-defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix given by: for each pair (i,j). For example: Matrix multiplication has the following properties: (AB)C = A(BC) for all k-by-m matrices A, m-by-n matrices B and n-by-p matrices C ("associativity"). (A+B)C = AC+BC for all m-by-n matrices A and B and n-by-k matrices C ("right distributivity"). C(A+B) = CA+CB for all m-by-n matrices A and B and k-by-m matrices C ("left distributivity"). Matrix multiplication is not commutative; that is, given matrices A and B and their product defined, then generally AB BA. It may also happen that AB is defined but BA is not defined. Besides the ordinary matrix multiplication just described, there exist other operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product.