# Matrices/K/Introduction/Section

A system of linear equations can easily be written with a matrix. This allows us to make the manipulations that lead to the solution of such a system without writing down the variables. Matrices are quite simple objects; however, they can represent quite different mathematical objects (e.g., a family of column vectors, a family of row vectors, a linear mapping, a table of physical interactions, a relation, a linear vector field, etc.), which one has to keep in mind in order to prevent wrong conclusions.

## Definition

Let $K$ denote a field, and let $I$ and $J$ denote index sets. An $I\times J$-matrix is a mapping
    $I\times J\longrightarrow K,(i,j)\longmapsto a_{ij}.$
If $I=\{1,\ldots ,m\}$ and $J=\{1,\ldots ,n\}$, then we talk about an $m\times n$-matrix. In this case, the matrix is usually written as
    ${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}.$
We will usually restrict to this last situation.
For every $i\in I$, the family $a_{ij}$ , $j\in J$, is called the $i$-th row of the matrix, which is usually written as a row tuple
(or row vector)
    $(a_{i1},a_{i2},\ldots ,a_{in}).$
For every $j\in J$, the family $a_{ij}$ , $i\in I$, is called the $j$-th column of the matrix, usually written as a column tuple (or column vector)
    ${\begin{pmatrix}a_{1j}\\a_{2j}\\\vdots \\a_{mj}\end{pmatrix}}.$
The elements $a_{ij}$ are called the entries of the matrix. For $a_{ij}$, the number $i$ is called the row index, and $j$ is called the column index of the entry. The position of the entry $a_{ij}$ is where the $i$-th row meets the $j$-th column. A matrix with $m=n$ is called a square matrix. An $m\times 1$-matrix is simply a column tuple (or column vector) of length $m$, and an $1\times n$-matrix is simply a row tuple (or row vector) of length $n$. The set of all matrices with $m$ rows and $n$ columns (and with entries in $K$) is denoted by $\operatorname {Mat} _{m\times n}(K)$, in case $m=n$ we also write $\operatorname {Mat} _{n}(K)$.
Two matrices $A,B\in \operatorname {Mat} _{m\times n}(K)$ are added by adding corresponding entries. The multiplication of a matrix $A$ with an element $r\in K$ (a scalar) is also defined entrywise, so
    ${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}+{\begin{pmatrix}b_{11}&b_{12}&\ldots &b_{1n}\\b_{21}&b_{22}&\ldots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\ldots &b_{mn}\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\ldots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\ldots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\ldots &a_{mn}+b_{mn}\end{pmatrix}}\,$
and
    $r{\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}={\begin{pmatrix}ra_{11}&ra_{12}&\ldots &ra_{1n}\\ra_{21}&ra_{22}&\ldots &ra_{2n}\\\vdots &\vdots &\ddots &\vdots \\ra_{m1}&ra_{m2}&\ldots &ra_{mn}\end{pmatrix}}\,.$
The multiplication of matrices is defined in the following way:

## Definition

Let $K$ denote a field, and let $A$ denote an $m\times n$-matrix and $B$ an $n\times p$-matrix over $K$. Then the matrix product
    $AB$
is the $m\times p$-matrix, whose entries are given by
    $c_{ik}=\sum _{j=1}^{n}a_{ij}b_{jk}\,.$
A matrix multiplication is only possible when the number of columns of the left-hand matrix equals the number of rows of the right-hand matrix. Just think of the scheme
    $(ROWROW){\begin{pmatrix}C\\O\\L\\U\\M\\N\end{pmatrix}}=(RC+O^{2}+WL+RU+OM+WN)\,,$
the result is an $1\times 1$-Matrix. In particular, one can multiply an $m\times n$-matrix $A$ with a column vector of length $n$ (the vector on the right), and the result is a column vector of length $m$. The two matrices can also be multiplied with roles interchanged,
    ${\begin{pmatrix}C\\O\\L\\U\\M\\N\end{pmatrix}}(ROWROW)={\begin{pmatrix}CR&CO&CW&CR&CO&CW\\OR&O^{2}&OW&OR&O^{2}&OW\\LR&LO&LW&LR&LO&LW\\UR&UO&UW&UR&UO&UW\\MR&MO&MW&MR&MO&MW\\NR&NO&NW&NR&NO&NW\end{pmatrix}}\,.$