Definition of m by n matrix

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