A matrix is a rectangular array of entries arranged in rows and columns. For example,

$$A=\begin{pmatrix} 1&2&3\ 4&5&6 \end{pmatrix}$$

has two rows and three columns, so it is a $2\times3$ matrix.

Entries and dimensions

The entry in row $i$ and column $j$ is written $A_{ij}$. In the matrix above,

$$A_{23}=6.$$

A matrix with the same number of rows and columns is square. A matrix with one row is a row matrix, while one with one column is a column matrix.

Vectors as columns

A vector can be written as a column matrix:

$$\mathbf v=\begin{pmatrix}v_1\v_2\v_3\end{pmatrix}.$$

This notation lets matrices act naturally on vectors later.

What matrices represent

A matrix is more than a storage table. Depending on context, its entries may represent the coefficients of a linear system, a transformation of coordinates, a collection of data or another structured linear relationship.

The same matrix arithmetic can then be used across these different interpretations.