Determine Order of Matrix

What is the Order of a Matrix?

The order of a matrix is a way to describe its size using the number of rows and columns. In mathematics, every matrix is written in a specific format, and its order helps us understand the structure clearly. We will discuss about the order of matrix in detail below:

Definition of Matrix Order

The order of a matrix is defined as the number of rows and columns it contains. If a matrix has $m$ rows and $n$ columns, then its order is written as:

$Order of Matrix = m \times n$

For example, if a matrix $A$ has 2 rows and 3 columns, its order will be $2 \times 3$.

Examples:

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

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, and 5, lie in the first column, 4,4 in the second column, and -3, 6 in the third column. Therefore, the order of a matrix is 2 x 3

Similarly, the Second matrix has order 3 x 2

and the third matrix has order 4 x 1

Representation of Order in m × n Form

The general way to represent a matrix order is:

$A = [a_{ij}]_{m \times n}$

A general $m \times n$ matrix is written as:

$\left[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{array}\right]$

This can be expressed in a more compact form as:

$\left[a_{ij}\right]_{m \times n}$

Here, $a_{ij}$ represents the element of the $i^{th}$ row and $j^{th}$ column, where

$i = 1,2,\ldots,m \quad ; \quad j = 1,2,\ldots,n$

For example, in matrix $A$ below, the element in row $2$ and column $3$ is $a_{23}$:

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

Note: A matrix is a systematic arrangement of symbols, numbers, or objects in rows and columns. It does not have a numerical value by itself. Usually, a matrix is denoted by capital letters such as $A, B, C$, etc.

Here:

• $i$ represents the row number ($1 \leq i \leq m$)

• $j$ represents the column number ($1 \leq j \leq n$)

• $m$ = number of rows

• $n$ = number of columns

For example:

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$

This matrix has $m = 2$ rows and $n = 3$ columns, so its order is $2 \times 3$.

Importance of Knowing the Order of a Matrix

Understanding the order of a matrix is essential because:

• Two matrices can be added or subtracted only if they have the same order.

• Matrix multiplication rules depend on the order of matrices.

• The existence of a determinant or inverse is possible only for square matrices (where $m = n$).

How to Determine the Order of a Matrix

To determine the order of a matrix, you need to count the number of rows and columns it has. The result is always written in the form $m \times n$, where $m$ = rows and $n$ = columns.

Step-by-Step Method to Find Order of Matrix

1. Write down the matrix clearly.

2. Count the number of horizontal rows.

3. Count the number of vertical columns.

4. Express the order as $m \times n$.

For example, if a matrix has 4 rows and 2 columns, its order is $4 \times 2$.

Examples of 2 × 2, 3 × 3, and Rectangular Matrices

2 × 2 Matrix

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

Here, $m = 2$, $n = 2$. Order = $2 \times 2$.

3 × 3 Matrix

$B = \begin{bmatrix} 1 & 0 & -1 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{bmatrix}$

Here, $m = 3$, $n = 3$. Order = $3 \times 3$.

Rectangular Matrix (3 × 4)

$C = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \end{bmatrix}$

Here, $m = 3$, $n = 4$. Order = $3 \times 4$.

Quick Tricks to Identify Rows and Columns

• Rows : Horizontal (count lines across).

• Columns : Vertical (count lines downward).

• Always write rows first, columns second.

• Order is never written as $n \times m$ by mistake.

Difference Between Order of a Matrix and Size of a Matrix

Students often confuse matrix order with matrix size. Let’s clarify.

Matrix Order vs. Number of Elements

• Order = number of rows × number of columns.

• Size = total number of elements.

Formula:

$Size = m \times n$

Example: If a matrix is of order $3 \times 4$, then size = $3 \times 4 = 12$ elements.

Common Mistakes Students Make

• Writing columns first instead of rows (e.g., $n \times m$ instead of $m \times n$).

• Confusing order with number of elements.

• Forgetting that $2 \times 3$ and $3 \times 2$ are different matrices.

Order of Special Types of Matrices

Different types of matrices have specific orders, and questions on this are common in competitive exams and board exams.

Order of Square Matrix

A square matrix has equal rows and columns.
Order: $n \times n$

Example:

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ → Order $2 \times 2$

Order of Row Matrix

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 $\mathrm{m}=1$.
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}}
$

Example: $\left[\begin{array}{llll}1 & 32 & 81 & -32\end{array}\right]$ has only 1 row. It has order $1 \times 4$

$\begin{bmatrix} 5 & 6 & 7 & 8 \end{bmatrix}$ → Order $1 \times 4$

Order of Column Matrix

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 $\mathrm{n}=1$.
Order: $m \times 1$

It is denoted by
$
\begin{aligned}
& {\left[\begin{array}{c}
a_{11} \\
a_{21} \\
a_{31} \\
\ldots \\
\ldots \\
a_{m 1}
\end{array}\right]_{\mathrm{m} \times 1}} \\
& \text { Example, }\left[\begin{array}{c}
2 \\
32 \\
3 \\
7
\end{array}\right]
\end{aligned}
$

This matrix has order 4 x 1

$\begin{bmatrix} 2 \\ 4 \\ 6 \\ 8 \end{bmatrix}$ → Order $4 \times 1$

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

Order of Zero Matrix and Identity Matrix

Zero Matrix: Can be of any order $m \times n$, but all elements are zero.
Example: $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ → Order $2 \times 2$

Identity Matrix: Always a square matrix ($n \times n$), with 1’s on the diagonal and 0’s elsewhere.
Example: $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ → Order $3 \times 3$

Properties of the order of a Matrix

1. In any matrix, the number of rows is always represented by the first number in the sequence and the number of columns by the second number.
2. Any two matrices can be added to or subtracted from one another as long as their orders are the same.
3. Any two matrices can only be multiplied if the number of rows in the second matrix equals the number of columns in the first matrix.
4. When a matrix has the order "m × n," its transpose matrix will have the order "n × m." Transpose matrices are created by turning a matrix's rows into columns and its columns into rows.

Solved Examples Based On Order Of Matrix

Example 1: The given matrix $\left[\begin{array}{l}3 \\ 8 \\ 9\end{array}\right]$ is

1) Row matrix of order 1x3

2) Row matrix of order 3x1

3) Column matrix of order 1x3

4) Column matrix of order 3x1

Solution:

As we have understood, there are 3 rows and only one column in the given matrix, i.e. m=3 and n=1, therefore, the order of the given matrix is 3x1.

Hence, the answer is the option 4.

Example 2: If matrix $A= \begin{bmatrix} 1\\ 3\\ 4 \end{bmatrix}$ and $B = \begin{bmatrix} -1\\ 0\\ 5 \end{bmatrix}$; then find C such that $A-C = B$.

Solution:

Column Matrix:

Single-column matrix order is $m \times 1$

- wherein

eg. $\begin{bmatrix} 7\\ -2\\ 1 \end{bmatrix}$

Let $C = \begin{bmatrix} a\\ b\\ c \end{bmatrix}$

Thus, $\begin{bmatrix} 1\\ 3\\ 4 \end{bmatrix}-\begin{bmatrix} a\\ b\\ c \end{bmatrix}= \begin{bmatrix} -1\\ 0\\ 5 \end{bmatrix}$

$1-a= -1, \quad 3-b = 0, \quad 4-c= 5$

$a=2, \quad b= 3, \quad c= -1$

$\begin{bmatrix} 2\\ 3\\ -1 \end{bmatrix}$

Example 3: The given matrix \left[\begin{array}{lll}{1} & {2} & {3}\end{array}\right] is a

1) Row matrix of order 1x3

2) Row matrix of order 3x1

3) Column matrix of order 1x3

4) Column matrix of order 3x1

Solution:

The given matrix has one row and three columns, i.e. m=1 and n=3.

Therefore the given matrix is of order 1x3.

Hence, the answer is the option 1.
Example 4: If the given matrix $\mathrm{A}$ is of order $5 \times 4$ and the element $a_{i j}=i^2+j^2$, then which of the following is true?
1) $a_{23}+a_{11}=a_{34}$
2) $a_{36}=a_{63}$
3) $a_{21}+a_{23}=a_{33}$
4) $a_{12}+a_{32}=a_{44}$

Solution:
$
\begin{aligned}
& a_{21}=2^2+1^2=5 \\
& a_{23}=2^2+3^2=13 \\
& a_{33}=3^2+3^2=18
\end{aligned}
$

Thus $a_{33}=a_{23}+a_{21}$
Hence, the answer is option 3.

Example 5: If matrix $A = \begin{bmatrix} 5 &7 &9 &11 \end{bmatrix}$, then find a matrix B that should be added to get $ C = \begin{bmatrix} 8 & 8 & 10 & 11 \end{bmatrix} $.

Solution:

Row Matrix -

Single-row matrix order is 1 \times n

- wherein

$ \begin{bmatrix} 1 & 2 & 3 & 7 \end{bmatrix} $

We can only add matrices in the same order so,

$A = \begin{bmatrix} 5 &7 & 9 &11\end{bmatrix}$, then $B= \begin{bmatrix} 3 &1 &1 &0 \end{bmatrix}$

List of topics related to the Order of Matrix

The order of a matrix is a basic yet important concept in linear algebra and competitive exams. To master it, students should also study related topics like types of matrices, matrix operations, determinants, and inverse of matrices. Here is a list of key topics connected with the order of matrix that will strengthen your preparation.

NCERT Resources

The NCERT resources for Class 12 Maths Chapter 3 – Matrices provide notes, solutions, and exemplar problems to understand matrix order, types, and operations easily.

NCERT Class 12 Maths Notes for Chapter 3 - Matrices

NCERT Solutions for Class 12 Maths for Chapter 3 - Matrices

NCERT Exemplar Class 12 Maths Solutions for Chapter 3 - Matrices

Practice Questions based on the Order of Matrix

Practicing questions on the order of a matrix builds a clear understanding of rows and columns, improves problem-solving speed, and helps in scoring well in exams.

Determine The Order Of Matrix - Practice Question MCQ

You can practice questions on the related topics using the links shared below: