Definition of matrix

Definition: A matrix is a rectangular array of numbers (or symbols, or functions) arranged in rows and columns.

Formally, an $m \times n$ matrix $A$ over a field $\mathbb{F}$ is a function $A: \{1,\ldots,m\} \times \{1,\ldots,n\} \to \mathbb{F}$, typically written as:

$$A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$$

Example:
$$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$$
is a $2 \times 3$ matrix with entries from $\mathbb{R}$.