$A = (a_{ij})$ in the matrix definition I have the following matrix definition

An m × n (read “m by n”) matrix A over a set S is a rectangular array
of elements of S arranged into m rows and n columns: (an mn matrix
shown)
We write $A = (a_{ij})$.

What is the meaning of $A = (a_{ij})$? $a_{ij}$ is an elements in the matrix, what's the point it writing this equality?
 A: Some times you want to talk about the matrix as a whole. Then you use $A$. Some times you want to talk about the elements. Then you use $a_{ij}$. The point of writing the equality is to formally establish that they are, ultimately, just two different notations for the same thing.
A: $(a_{ij})_{1\leq i\leq m,1\leq j\leq n}$
This is how we write double sequences. And a matrix is a finite  double sequence. So matrix $A$ is written as
$$A=(a_{ij})_{1\leq i\leq m,1\leq j\leq n}$$
But when from the context, size of the matrix is clear then mostly we omit the suffix and simply we write
$$A=(a_{ij})$$
A: It means: we call the matrix $A$ and the element in the $i$th row and $j$th column $a_{ij}$. Furthermore, we sometimes don't want to introduce a new symbol for the matrix formed by some elements $a_{ij}$. Then, we simply write $(a_{ij})$ in the implicit understanding that this denotes the matrix formed by the different $a_{ij}$ and not a specific $a_{ij}$ enclosed by normal brackets.
Furthermore it means: we will do the same for every matrix. Thus, from now on whenever some upper case letter denotes a matrix, we will denote by a corresponding lower case letter with two indices the elements of the matrix.
A: By

We write $A=(a_{ij})$

the author means to say the following:

If at any point from here on, we write $a_{ij}$, we are referring to the value in the $i$-th row and $j$-th column of matrix $A$.

In other words, a little counterintuitively, the expression $A=(a_{ij})$ is not actually defininig the value $A$, but rather the values $a_{ij}$ for $1\leq i\leq m$ and $1\leq j\leq n$.
