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Matrices are useful to record data that depends on two categories, and to keep track of the coefficients of systems of linear equations and linear transformations.
The term is also used in other areas, see matrix.

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 mbyn matrix (or m×n matrix) and m and n are called its dimensions. For example the matrix below is a 4by3 matrix:
The entry of a matrix A that lies in the ith row and the jth column is called the i,jentry or (i,j)th entry of A.
This is written as A[i,j] or A_{i,j}, or in notation of the C programming language, A[i][j]
. In the example above, A[2,3]=7.
The notation A = (a_{ij}) means that A[i,j] = a_{ij} for all indices i and j.
Adding and multiplying matrices
If two mbyn matrices A and B are given, we may define their sum A + B as the mbyn matrix computed by adding corresponding elements, i.e., (A + B)[i, j] = A[i, j] + B[i, j]. For example
\begin{bmatrix} 1 & 3 & 2 \\ 1 & 0 & 0 \\ 1 & 2 & 2 \end{bmatrix}+
\begin{bmatrix} 0 & 0 & 5 \\ 7 & 5 & 0 \\ 2 & 1 & 1 \end{bmatrix}
\begin{bmatrix} 1 & 3 & 7 \\ 8 & 5 & 0 \\ 3 & 3 & 3 \end{bmatrix}</math>
Another, much less often used notion of matrix addition can be found at Direct sum (Matrix).
If a matrix A and a number c are given, we may define the scalar multiplication cA by (cA)[i, j] = cA[i, j]. For example
\begin{bmatrix} 1 & 8 & 3 \\ 4 & 2 & 5 \end{bmatrix}
\begin{bmatrix} 2 & 16 & 6 \\ 8 & 4 & 10 \end{bmatrix}</math>
These two operations turn the set M(m, n, R) of all mbyn matrices with real entries into a real vector space of dimension mn.
Multiplication of two matrices is welldefined only if the number of columns of the first matrix is the same as the number of rows of the second matrix. If A is an mbyn matrix (m rows, n columns) and B is an nbyp matrix (n rows, p columns), then their product AB is the mbyp matrix (m rows, p columns) given by
\begin{bmatrix} 1 & 0 & 2 \\ 1 & 3 & 1 \\ \end{bmatrix}\times
\begin{bmatrix} 3 & 1 \\ 2 & 1 \\ 1 & 0 \end{bmatrix}
\begin{bmatrix} 5 & 1 \\ 4 & 2 \\ \end{bmatrix}</math>
This multiplication has the following properties:
For other, less commonly encountered ways to multiply matrices, see matrix multiplication.
Linear transformations, Ranks and Transpose
Matrices can conveniently represent linear transformations because matrix multiplication neatly corresponds to the composition of maps, as will be described next.
Here and in the sequel we identify R^{n} with the set of "rows" or nby1 matrices. For every linear map f : R^{n} > R^{m} there exists a unique mbyn matrix A such that f(x) = Ax for all x in R^{n}. We say that the matrix A "represents" the linear map f. Now if the kbym matrix B represents another linear map g : R^{m} > R^{k}, then the linear map g o f is represented by BA. This follows from the abovementioned associativity of matrix multiplication.
The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.
The transpose of an mbyn matrix A is the nbym matrix A^{tr} (also sometimes written as A^{T} or ^{t}A) gotten by turning rows into columns and columns into rows, i.e. A^{tr}[i, j] = A[j, i] for all indices i and j. If A describes a linear map with respect to two bases, then the matrix A^{tr} describes the transpose of the linear map with respect to the dual bases, see dual space.
We have (A + B)^{tr} = A^{tr} + B^{tr} and (AB)^{tr} = B^{tr} * A^{tr}.
Square matrices and Related definitions
A square matrix is a matrix which has the same number of rows as columns. The set of all square nbyn matrices, together with matrix addition and matrix multiplication is a ring. Unless n = 1, this ring is not commutative.
M(n, R) , the ring of real square matrices, is a real unitary associative algebra. M(n, C), the ring of complex square matrices, is a complex associative algebra.
The unit matrix or identity matrix I_{n}, with elements on the main diagonal set to 1 and all other elements set to 0, satisfies MI_{n}=M and I_{n}N=N for any mbyn matrix M and nbyk matrix N. For example, if n = 3:
I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}</math> The identity matrix is the identity element in the ring of square matrices.
Invertible elements in this ring are called invertible matrices or nonsingular matrices. An n by n matrix A is invertible if and only if there exists a matrix B such that
If λ is a number and v is a nonzero vector such that Av = λv, then we call v an eigenvector of A and γ the associated eigenvalue. The number λ is an eigenvalue of A if and only if AλI_{n} is not invertible, which happens if and only if p_{A}(λ) = 0. Here p_{A}(x) is the characteristic polynomial of A. This is a polynomial of degree n and has therefore n complex roots (counting multiple roots according to their multiplicity). In this sense, every square matrix has n complex eigenvalues.
The determinant of a square matrix A is the product of its n eigenvalues, but it can also be defined by the Leibniz formula. Invertible matrices are precisely those matrices with nonzero determinant.
The GaussJordan elimination algorithm is of central importance: it can be used to compute determinants, ranks and inverses of matrices and to solve systems of linear equations.
The trace of a square matrix is the sum of its diagonal entries, which equals the sum of its n eigenvalues.
A Partitioned Matrix or Block Matrix is a matrix of matricies. For example, take a matrix P:
We could partition it into a 2by2 partitioned matrix like this:
This technique is used to cut down calculations of matricies, columnrow expansions, and many computer science applications, including VLSI chip design.
Classes of real and complex matrices
Certain special matrices are so important that they are given special names, as listed in list of matrices. Below are some examples:
Matrices with entries in arbitrary rings
If we start with a ring R, we can consider the set M(m,n, R) of all m by n matrices with entries in R. Addition and multiplication of these matrices can be defined as above, and it has the same properties. The set M(n, R) of all square n by n matrices over R is a ring in its own right, isomorphic to the endomorphism ring of the left R module R^{n}.
If R is commutative, then M(n, R) is a unitary associative algebra over R. It is then also meaningful to define the determinant of square matrices using the Leibniz formula; a matrix is invertible if and only if its determinant is invertible in R.
All statements mentioned above for real or complex matrices remain correct for matrices over an arbitrary field.
Matrices over a polynomial ring are important in the study of control theory.
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