In mathematics, a matrix (plural: matrices) is a rectangle Equivalently, . of , symbols, or expressions, arranged in and . For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns:
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to manipulate 3D models and project them onto a 2dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.K. Bryan and T. Leise. The $25,000,000,000 eigenvector: The linear algebra behind Google. SIAM Review, 48(3):569–581, 2006. Matrix calculus generalizes classical analytical notions such as and exponentials to higher dimensions. Matrices are used in economics to describe systems of economic relationships.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as Sparse matrix and diagonal matrix, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.
1.3 & 0.6 \\ 20.4 & 5.5 \\ 9.7 & 6.2 \end{bmatrix}.
The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.
Matrices with a single row are called , and those with a single column are called column vectors. A matrix with the same number of rows and columns is called a square matrix. A matrix with an infinite number of rows or columns (or both) is called an infinite matrix. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
$\backslash begin\{bmatrix\}3\; \&\; 7\; \&\; 2\; \backslash end\{bmatrix\}$  A matrix with one row, sometimes used to represent a vector 
$\backslash begin\{bmatrix\}4\; \backslash \backslash \; 1\; \backslash \backslash \; 8\; \backslash end\{bmatrix\}$  A matrix with one column, sometimes used to represent a vector 
$\backslash begin\{bmatrix\}$9 & 13 & 5 \\ 1 & 11 & 7 \\ 2 & 6 & 3 \end{bmatrix}  A matrix with the same number of rows and columns, sometimes used to represent a linear transformation from a vector space to itself, such as reflection, rotation, or Shear mapping. 
\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} =\left( \begin{array}{rrrr}
a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array} \right) =\left(a_{ij}\right) \in \mathbb{R}^{m \times n}.
The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are usually symbolized using uppercase letters (such as A in the examples above), while the corresponding lowercase letters, with two subscript indices (for example, a_{11}, or a_{1,1}), represent the entries. In addition to using uppercase letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (nonitalic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a doubleunderline with the variable name, with or without boldface style, (for example, $\backslash underline\{\backslash underline\{A\}\}$).
The entry in the ith row and jth column of a matrix A is sometimes referred to as the i, j, ( i, j), or ( i, j)^{th} entry of the matrix, and most commonly denoted as a_{ i, j}, or a_{ ij}. Alternative notations for that entry are A i,j or A_{ i,j}. For example, the (1,3) entry of the following matrix A is 5 (also denoted a_{13}, a_{1,3}, A1,3 or A_{ 1,3}):
\mathbf{A}=\begin{bmatrix} 4 & 7 & \color{red}{5} & 0 \\ 2 & 0 & 11 & 8 \\ 19 & 1 & 3 & 12 \end{bmatrix}
Sometimes, the entries of a matrix can be defined by a formula such as a_{ i, j} = f( i, j). For example, each of the entries of the following matrix A is determined by a_{ ij} = i − j.
Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m× n matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an mby n matrix are indexed by and . This article follows the more common convention in mathematical writing where enumeration starts from 1.
An asterisk is occasionally used to refer to whole rows or columns in a matrix. For example, a_{ i,∗} refers to the i^{th} row of A, and a_{∗, j} refers to the j^{th} column of A. The set of all mby n matrices is denoted 𝕄( m, n).
Matrix addition  The sum A+ B of two mby n matrices A and B is calculated entrywise:
 $$ \begin{bmatrix} 1 & 3 & 1 \\ 1 & 0 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 5 \\ 7 & 5 & 0 \end{bmatrix} = \begin{bmatrix} 1+0 & 3+0 & 1+5 \\ 1+7 & 0+5 & 0+0 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 6 \\ 8 & 5 & 0 \end{bmatrix} 
Scalar multiplication  The product c A of a number c (also called a scalar in the parlance of abstract algebra) and a matrix A is computed by multiplying every entry of A by c:
 $2\; \backslash cdot$ \begin{bmatrix} 1 & 8 & 3 \\ 4 & 2 & 5 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 & 2\cdot 8 & 2\cdot 3 \\ 2\cdot 4 & 2\cdot 2 & 2\cdot 5 \end{bmatrix} = \begin{bmatrix} 2 & 16 & 6 \\ 8 & 4 & 10 \end{bmatrix} 
Transpose  The transpose of an mby n matrix A is the nby m matrix A^{T} (also denoted A^{tr} or ^{t} A) formed by turning rows into columns and vice versa:
 $$ \begin{bmatrix} 1 & 2 & 3 \\ 0 & 6 & 7 \end{bmatrix}^\mathrm{T} = \begin{bmatrix} 1 & 0 \\ 2 & 6 \\ 3 & 7 \end{bmatrix} 
Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, that is, the matrix sum does not depend on the order of the summands: A + B = B + A. The transpose is compatible with addition and scalar multiplication, as expressed by ( c A)^{T} = c( A^{T}) and ( A + B)^{T} = A^{T} + B^{T}. Finally, ( A^{T})^{T} = A.
where 1 ≤ i ≤ m and 1 ≤ j ≤ p. For example, the underlined entry 2340 in the product is calculated as
\begin{bmatrix} 0 & \underline{1000} \\ 1 & \underline{100} \\ 0 & \underline{10} \\ \end{bmatrix} &= \begin{bmatrix} 3 & \underline{2340} \\ 0 & 1000 \\ \end{bmatrix}. \end{align}
Matrix multiplication satisfies the rules ( AB) C = A( BC) (associativity), and ( A+ B) C = AC+ BC as well as C( A+ B) = CA+ CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are mby n and nby k matrices, respectively, and Even if both products are defined, they need not be equal, that is, generally
\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}= \begin{bmatrix} 0 & 1\\ 0 & 3\\ \end{bmatrix}, whereas
\begin{bmatrix} 1 & 2\\ 3 & 4\\ \end{bmatrix}= \begin{bmatrix} 3 & 4\\ 0 & 0\\ \end{bmatrix} .
Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product. They arise in solving matrix equations such as the Sylvester equation.
\mathbf{A}=\begin{bmatrix} 1 & \color{red}{2} & 3 & 4 \\ 5 & \color{red}{6} & 7 & 8 \\ \color{red}{9} & \color{red}{10} & \color{red}{11} & \color{red}{12} \end{bmatrix} \rightarrow \begin{bmatrix} 1 & 3 & 4 \\ 5 & 7 & 8 \end{bmatrix}.
The minors and cofactors of a matrix are found by computing the determinant of certain submatrices.
A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain... Other authors define a principal submatrix as one in which the first k rows and columns, for some number k, are the ones that remain;. this type of submatrix has also been called a leading principal submatrix..
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, 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.
For example, the 2×2 matrix
can be viewed as the transform of the unit square into a parallelogram with vertices at , , , and . The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors $\backslash begin\{bmatrix\}\; 0\; \backslash \backslash \; 0\; \backslash end\{bmatrix\},\; \backslash begin\{bmatrix\}\; 1\; \backslash \backslash \; 0\; \backslash end\{bmatrix\},\; \backslash begin\{bmatrix\}\; 1\; \backslash \backslash \; 1\; \backslash end\{bmatrix\}$ and $\backslash begin\{bmatrix\}0\; \backslash \backslash \; 1\backslash end\{bmatrix\}$ in turn. These vectors define the vertices of the unit square.
The following table shows a number of 2by2 matrices with the associated linear maps of R^{2}. The blue original is mapped to the green grid and shapes. The origin (0,0) is marked with a black point.
Shear mapping with m=1.25.  Reflection through the vertical axis  Squeeze mapping with r=3/2  Scaling by a factor of 3/2  Rotation matrix by π/6^{R} = 30° 
$\backslash begin\{bmatrix\}\; 1\; \&\; 1.25\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\{bmatrix\}$  $\backslash begin\{bmatrix\}\; 1\; \&\; 0\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\{bmatrix\}$  $\backslash begin\{bmatrix\}\; 3/2\; \&\; 0\; \backslash \backslash \; 0\; \&\; 2/3\; \backslash end\{bmatrix\}$  $\backslash begin\{bmatrix\}\; 3/2\; \&\; 0\; \backslash \backslash \; 0\; \&\; 3/2\; \backslash end\{bmatrix\}$  $\backslash begin\{bmatrix\}\backslash cos(\backslash pi\; /\; 6^\{R\})\; \&\; \backslash sin(\backslash pi\; /\; 6^\{R\})\backslash \backslash \; \backslash sin(\backslash pi\; /\; 6^\{R\})\; \&\; \backslash cos(\backslash pi\; /\; 6^\{R\})\backslash end\{bmatrix\}$ 
Under the bijection between matrices and linear maps, matrix multiplication corresponds to composition of maps: if a kby m matrix B represents another linear map g : R^{ m} → R^{ k}, then the composition is represented by BA since
The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. Equivalently it is the Hamel dimension of the image of the linear map represented by A. The rank–nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix.
$$\begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \\ \end{bmatrix} 
$$\begin{bmatrix} a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix} 
$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \\ \end{bmatrix} 
1 & 0 \\ 0 & 1 \end{bmatrix},\ \cdots ,\ I_n = \begin{bmatrix}
1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix}It is a square matrix of order n, and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:
A nonzero scalar multiple of an identity matrix is called a scalar matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.
By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. This theorem can be generalized to infinitedimensional situations related to matrices with infinitely many rows and columns, see below.
$\backslash begin\{bmatrix\}$1/4 & 0 \\ 0 & 1 \\ \end{bmatrix}  $\backslash begin\{bmatrix\}$1/4 & 0 \\ 0 & 1/4 \end{bmatrix} 
Q( x, y) = 1/4 x^{2} + y^{2}  Q( x, y) = 1/4 x^{2} − 1/4 y^{2} 
Points such that Q( x, y)=1 (Ellipse).  Points such that Q( x, y)=1 (Hyperbola). 
A symmetric matrix is positivedefinite if and only if all its eigenvalues are positive, that is, the matrix is positivesemidefinite and it is invertible. The table at the right shows two possibilities for 2by2 matrices.
Allowing as input two different vectors instead yields the bilinear form associated to A:
An orthogonal matrix A is necessarily invertible (with inverse ), Unitary matrix (), and Normal matrix (). The determinant of any orthogonal matrix is either +1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear transformation, every orthogonal matrix with determinant +1 is a pure rotation, while every orthogonal matrix with determinant 1 is either a pure reflection, or a composition of reflection and rotation.
The complex number analogue of an orthogonal matrix is a unitary matrix.
The determinant of 2by2 matrices is given by
The determinant of a product of square matrices equals the product of their determinants:
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1. Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, that is, determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1by1 matrix, which is its unique entry, or even the determinant of a 0by0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.
To choose the most appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called numerical linear algebra. As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability.
Determining the complexity of an algorithm means finding or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, multiplication of matrices. For example, calculating the matrix product of two nby n matrix using the definition given above needs n^{3} multiplications, since for any of the n^{2} entries of the product, n multiplications are necessary. The Strassen algorithm outperforms this "naive" algorithm; it needs only n^{2.807} multiplications. A refined approach also incorporates specific features of the computing devices.
In many practical situations additional information about the matrices involved is known. An important case are sparse matrix, that is, matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.
An algorithm is, roughly speaking, numerically stable, if little deviations in the input values do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace expansion (Adj ( A) denotes the adjugate matrix of A)
Although most computer languages are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.For example, Mathematica, see
The LU decomposition factors matrices as a product of lower ( L) and an upper triangular matrices ( U). Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row. Singular value decomposition expresses any matrix A as a product UDV^{∗}, where U and V are unitary matrix and D is a diagonal matrix.
The eigendecomposition or diagonalization expresses A as a product VDV^{−1}, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizable. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form, that is to say matrices whose only nonzero entries are the eigenvalues λ_{1} to λ_{n} of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right. Given the eigendecomposition, the n^{th} power of A (that is, nfold iterated matrix multiplication) can be calculated via
More generally, matrices with entries in a ring R are widely used in mathematics. Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M( n, R) of all square nby n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left Rmodule R^{ n}. If the ring R is commutative ring, that is, its multiplication is commutative, then M( n, R) is a unitary noncommutative (unless n = 1) associative algebra over R. The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible. Matrices over are called supermatrix.
Matrices do not always have all their entries in the same ring – or even in any ring at all. One special but common case is block matrix, which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any ring; but their sizes must fulfil certain compatibility conditions.
These properties can be restated in a more natural way: the category of all matrices with entries in a field $k$ with multiplication as composition is equivalent to the category of finite dimensional and linear maps over this field.
More generally, the set of m× n matrices can be used to represent the Rlinear maps between the free modules R^{ m} and R^{ n} for an arbitrary ring R with unity. When n = m composition of these maps is possible, and this gives rise to the matrix ring of n× n matrices representing the endomorphism ring of R^{ n}.
Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroup of (that is, a smaller group contained in) their general linear group, called a special linear group. Orthogonal matrices, determined by the condition
Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group. General groups can be studied using matrix groups, which are comparatively well understood, by means of representation theory.See any reference in representation theory or group representation.
If R is any ring with unity, then the ring of endomorphisms of $M=\backslash bigoplus\_\{i\backslash in\; I\}R$ as a right R module is isomorphic to the ring of column finite matrices $\backslash mathbb\{CFM\}\_I(R)$ whose entries are indexed by $I\backslash times\; I$, and whose columns each contain only finitely many nonzero entries. The endomorphisms of M considered as a left R module result in an analogous object, the row finite matrices $\backslash mathbb\{RFM\}\_I(R)$ whose rows each only have finitely many nonzero entries.
If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map f: V→ W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector v of coefficients, only finitely many entries v_{ i} are nonzero. Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of A however: in the product A· v there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover, this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns do. One also sees that products of two matrices of the given type is well defined (provided as usual that the columnindex and rowindex sets match), is again of the same type, and corresponds to the composition of linear maps.
If R is a normed ring, then the condition of row or column finiteness can be relaxed. With the norm in place, absolutely convergent series can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring.
In that vein, infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuity questions arise, which again results in certain constraints that must be imposed. However, the explicit point of view of matrices tends to obfuscate the matter,"Not much of matrix theory carries over to infinitedimensional spaces, and what does is not so useful, but it sometimes helps." and the abstract and more powerful tools of functional analysis can be used instead.
Complex numbers can be represented by particular real 2by2 matrices via
Early encryption techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break. Computer graphics uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrix to accomplish tasks such as projecting a threedimensional object onto a twodimensional screen, corresponding to a theoretical camera observation. Matrices over a polynomial ring are important in the study of control theory.
Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss chemical bond and spectroscopy. Examples are the overlap matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method.
Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: R^{ n} → R^{ m}. If f_{1}, ..., f_{ m} denote the components of f, then the Jacobi matrix is defined as
Partial differential equations can be classified by considering the matrix of coefficients of the highestorder differential operators of the equation. For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.
The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.. See also stiffness method.
Statistics also makes use of matrices in many different forms. Descriptive statistics is concerned with describing data sets, which can often be represented as data matrices, which may then be subjected to dimensionality reduction techniques. The covariance matrix encodes the mutual variance of several . Another technique using matrices are linear least squares, a method that approximates a finite set of pairs ( x_{1}, y_{1}), ( x_{2}, y_{2}), ..., ( x_{ N}, y_{ N}), by a linear function
Random matrix are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics.
Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in particle accelerators, where noninteracting particles head towards each other and collide in a small interaction zone, with a new set of noninteracting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the Smatrix, which encodes all information about the possible interactions between particles.
The behaviour of many Electronics components can be described using matrices. Let A be a 2dimensional vector with the component's input voltage v_{1} and input current i_{1} as its elements, and let B be a 2dimensional vector with the component's output voltage v_{2} and output current i_{2} as its elements. Then the behaviour of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedance element ( h_{12}), one admittance element ( h_{21}) and two dimensionless elements ( h_{11} and h_{22}). Calculating a circuit now reduces to multiplying matrices.
The term "matrix" (Latin for "womb", derived from '—mother) was coined by James Joseph Sylvester in 1850,Although many sources state that J. J. Sylvester coined the mathematical term "matrix" in 1848, Sylvester published nothing in 1848. (For proof that Sylvester published nothing in 1848, see: J. J. Sylvester with H. F. Baker, ed., The Collected Mathematical Papers of James Joseph Sylvester (Cambridge, England: Cambridge University Press, 1904), vol. 1.) His earliest use of the term "matrix" occurs in 1850 in: J. J. Sylvester (1850) "Additions to the articles in the September number of this journal, "On a new class of theorems," and on Pascal's theorem," The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 37''' : 363370. From page 369: "For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This does not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants … " who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:
Arthur Cayley published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done. Instead he defined operations such as addition, subtraction, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held true. Cayley investigated and demonstrated the noncommutative property of matrix multiplication as well as the commutative property of matrix addition. Early matrix theory had limited the use of arrays almost exclusively to determinants and Arthur Cayley's abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858 Arthur Cayley published his A memoir on the theory of matrices Phil.Trans. 1858, vol.148, pp.1737 Math. Papers II 475496 in which he proposed and demonstrated the Cayley–Hamilton theorem.
An English mathematician named Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation A = a_{ i, j} to represent a matrix where a_{ i, j} refers to the ith row and the jth column.
The modern study of determinants sprang from several sources. number theory problems led Gauss to relate coefficients of , that is, expressions such as and in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, are noncommutative. Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = a_{ i, j} the following: replace the powers a_{ j}^{ k} by a_{ jk} in the polynomial
Many theorems were first established for small matrices only, for example the Cayley–Hamilton theorem was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by Hamilton for 4×4 matrices. Georg Frobenius, working on , generalized the theorem to all dimensions (1898). Also at the end of the 19th century the Gauss–Jordan elimination (generalizing a special case now known as Gauss elimination) was established by Jordan. In the early 20th century, matrices attained a central role in linear algebra, partially due to their use in classification of the hypercomplex number systems of the previous century.
The inception of matrix mechanics by Heisenberg, Max Born and Pascual Jordan led to studying matrices with infinitely many rows and columns. Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as on , which, very roughly speaking, correspond to Euclidean space, but with an infinity of Hamel dimension.
Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) use the word "matrix" in the context of their axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the "bottom" (0 order) the function is identical to its extension:
Alfred Tarski in his 1946 Introduction to Logic used the word "matrix" synonymously with the notion of truth table as used in mathematical logic.Tarski, Alfred; (1946) Introduction to Logic and the Methodology of Deductive Sciences, Dover Publications, Inc, New York NY, .

