Notation for entry of matrix

Notation for entry of a matrix: The entry in row $i$ and column $j$ of a matrix $A$ is denoted by $a_{ij}$, $A_{ij}$, or $(A)_{ij}$.

The first subscript always refers to the row index, and the second to the column index.

Example: If
$$A = \begin{pmatrix} 3 & 7 & 1 \\ 5 & 2 & 9 \end{pmatrix}$$
then $a_{11} = 3$, $a_{12} = 7$, $a_{21} = 5$, and $a_{23} = 9$.

In compact notation, we may write $A = (a_{ij})$ to denote the entire matrix, where $1 \leq i \leq m$ and $1 \leq j \leq n$.