A **matrix** is a rectangular array of numbers. This example has $m$ rows and $n$ columns, so we say it is $m\times n$ $A=\left(\begin{array}{ccccc}a_{11} & a_{12} & a_{13} & \cdots & a_{1 n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2 n} \\ \vdots & \vdots & \vdots & \ddots& \vdots \\ a_{m 1} & a_{m 2} & a_{m 3} & \cdots & a_{m n}\end{array}\right)=(a_{ij})\begin{cases}i=1,\ldots,m\\j=1,\ldots,n\end{cases}$ - The third *column vector* $\mathbf a_{\star3}$ is $\left(\begin{array}{c}a_{13} \\ a_{23} \\ \vdots \\ a_{m 3}\end{array}\right)$ - An $m\times 1$ matrix - The second *row vector* $\mathbf a_{2\star}$ is $(a_{21}, a_{22}, a_{23}, \cdots, a_{2 n})$ - A $1\times n$ matrix - The *entry* in row $i$ and column $j$ is called the $ij$-entry, or $ij$-component $a_{ij}$ - A $1\times1$ matrix Matrices can be added according to [[matrix addition]] In [[Linear Algebra]], **matrices** act on [[Vector|vectors]] by [[matrix vector multiplication]], which is [[matrix multiplication]]. Special matrices [[square matrix]] [[row vector]] [[column vector]] [[zero matrix]]