Concept and Notation of a Matrix

What Is a Matrix?

In many real situations we need to record several numbers that are all related to each other in a grid-like way -- for instance, the marks scored by $3$ students in $4$ subjects, or the number of items of $2$ kinds sent from $3$ warehouses to $2$ shops. Writing such numbers in a rectangular arrangement, enclosed in a single pair of brackets, is far more convenient than listing them one by one in words. This rectangular arrangement is called a matrix.

Definition. A matrix is a rectangular array of numbers (or, more generally, of symbols) arranged in rows and columns, enclosed within a pair of square brackets $[\,\cdot\,]$ or round brackets $(\,\cdot\,)$. For example,

is a matrix with $2$ rows and $3$ columns. Each number inside the array is called an element or entry of the matrix.

Notation

A matrix is usually named by a single capital letter, such as $A$, $B$, $M$. If a matrix $A$ has $m$ rows and $n$ columns, it is written in general as

written compactly as $A=[a_{ij}]_{m\times n}$, or simply $A=[a_{ij}]$ when the order is understood from context. Here $a_{ij}$ denotes the entry that sits in the $i$th row and $j$th column of $A$ -- the row index $i$ always comes first, the column index $j$ second. For instance, in the matrix $A$ above, $a_{12}=-1$ (row $1$, column $2$) and $a_{23}=3$ (row $2$, column $3$).

Constructing a Matrix from a Formula

A common type of question gives a rule $a_{ij}=f(i,j)$ for the general entry and asks for the matrix itself. To build it, substitute every valid pair $(i,j)$ into the formula, keeping the row index $i$ fixed while the column index $j$ runs through its full range, then move to the next row. Example 1 and Exercise: Matrix Notation and Types, Q2 both use this construction.