Definition of size of a matrix

Definition: The size (or dimensions) of a matrix is the pair $(m, n)$, where $m$ is the number of rows and $n$ is the number of columns. We say the matrix has size $m \times n$ (read "$m$ by $n$").

The size determines the total number of entries: an $m \times n$ matrix has $mn$ entries.

Example:
$$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 2 & 3 \end{pmatrix}$$
has size $3 \times 2$ (3 rows, 2 columns).

A matrix with one row ($1 \times n$) is called a row vector. A matrix with one column ($m \times 1$) is called a column vector.