Order of a Matrix

The Meaning of "Order of a Matrix"

The order of a matrix describes its shape: how many rows and how many columns it has.

If a matrix has $m$ rows and $n$ columns, it is a matrix of order $m \times n$ (read "m by n"), also called an $m \times n$ matrix. The order is always written as rows × columns.

For example:

• Matrix $A$ has 3 rows and 2 columns → order $3 \times 2$.
• Matrix $B$ has 3 rows and 3 columns → order $3 \times 3$.
• Matrix $C$ has 2 rows and 3 columns → order $2 \times 3$.

The number of elements is the product of rows and columns, so $A$ and $C$ each have $6$ elements and $B$ has $9$.

The General Notation for a Matrix

In general, an $m \times n$ matrix is a rectangular array of elements arranged in $m$ rows and $n$ columns, written as:

Here:

• $i$ is the row index and $j$ is the column index.
• $a_{ij}$ is the element in the $i$-th row and $j$-th column, also called the $(i, j)$-th element of $A$.

The $i$-th row consists of $a_{i1}, a_{i2}, a_{i3}, \dots, a_{in}$, and the $j$-th column consists of $a_{1j}, a_{2j}, a_{3j}, \dots, a_{mj}$.

Representing Points and Figures Using Matrices

A point $(x, y)$ in a plane can be represented as a column or row matrix:

For example, the point $P(0, 1)$ can be written as $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ or $[0 \; 1]$.

This extends to the vertices of a closed rectilinear figure. For a quadrilateral $ABCD$ with vertices $A(1, 0)$, $B(3, 2)$, $C(1, 3)$, $D(-1, 2)$, the vertices can be stored either as a $2 \times 4$ matrix (each column a vertex) or a $4 \times 2$ matrix (each row a vertex): …