A rectangular arrangement of objects (numbers or symbols or any other objects) is called a matrix (plural: matrices).

Example:

1. $\left[\begin{array}{ccc}2 & 4 & -3 \\ 5 & 4 & 6\end{array}\right]$
2. $\left[\begin{array}{cc}2 & 4 i+3 \\ 5 & 4 \\ 3 i & -75\end{array}\right]$
$\left[\begin{array}{c}2 \\ -5 \\ 3 i \\ 71\end{array}\right]$

Order of Matrix

Rows and Columns:

The horizontal objects denote a row and the vertical ones denote a column.

Eg, in the first matrix above, elements 2, 4 and -3 lie in the first row and 5, 4 and 6 in the second row

Also, 2, 5, lie in the first column, 4,4 in the second column, and -3, and 6 in the third column

Order of a matrix: 

Matrix of order m × n, (read as m by n matrix) means that the matrix has m number of rows and n number of columns.

E.g.,

The first matrix has order 2 x 3

The second matrix has order 3 x 2

The third matrix has order 4 x 1

Representation of a m x n matrix:

$\left[\begin{array}{cccc}a_{11} & a_{12} & \ldots & a_{1 n} \\ a_{21} & a_{22} & \ldots & a_{2 n} \\ \ldots & \ldots & \ldots & \ldots \\ a_{m 1} & a_{m 2} & \ldots & a_{m n}\end{array}\right]$

This representation can be represented in a more compact form as  $\left[a_{i j}\right]_{m \times n}$

Where $a_{i j}$ represents the element of i^th row and j^th column and i = 1,2,...,m; j = 1,2,...,n.

For example, to locate the entry in matrix A identified as a_ij, we look for the entry in row i, column j. In matrix A, shown below, the entry in row 2, column 3 is a₂₃.

$A=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$

Note:

Matrix is only a representation of the symbol, number or object. It does not have any value. Usually, a matrix is denoted by capital letters.

Types of Matrices

Row matrix: A matrix containing only one row is called a row matrix. So a matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$  is said to be a row matrix when m = 1.

It can be denoted by

$\left[\begin{array}{llllll}a_{11} & a_{12} & a_{13} & \ldots & \ldots & a_{1 n}\end{array}\right]_{1 \times \mathrm{n}}$ 

Eg, 

[ 1    32    81    -32 ] has only 1 row. It has order 1 x 4

Column matrix: A matrix containing only one column is known as a column matrix. So a matrix  $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$  is said to be a column matrix when n = 1.

It is denoted by

$
\left[\begin{array}{c}
a_{11} \\
a_{21} \\
a_{31} \\
\cdots \\
\cdots \\
a_{m 1}
\end{array}\right]_{\mathrm{m} \times 1}
$
Eg,

$
\left[\begin{array}{c}
2 \\
32 \\
3 \\
7
\end{array}\right]
$

This matrix has order 4 x 1

Note:

A matrix that contains only one row or one column is also known as a vector i.e. row vectors and column vectors.