**Definition:** An **$m \times n$ matrix** is a matrix with exactly $m$ rows and $n$ columns.
It is written as:
$$A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}$$
where each $a_{ij}$ belongs to some field (typically $\mathbb{R}$ or $\mathbb{C}$).
**Example:** A $2 \times 3$ matrix:
$$\begin{pmatrix} 1 & -2 & 0 \\ 3 & 4 & 5 \end{pmatrix}$$
Special cases:
- When $m = n$, the matrix is **square**
- When $m = 1$, it is a **row vector**
- When $n = 1$, it is a **column vector**