Determine Order of Matrix

A matrix (plural: matrices) is a rectangular arrangement of symbols along rows and columns that might be real or complex numbers. Thus, a system of m x n symbols arranged in a rectangular formation along m rows and n columns is called an m by n matrix (which is written as m x n matrix). The order of the matrix helps to determine the number of rows and columns. The order helps us to understand the type of matrix and the total elements present in the matrix.

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In this article, we will cover the concept order of matrices. This category falls under the broader category of Matrices, which is a crucial Chapter in class 12 Mathematics. It is essential not only for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Questions based on this topic have been asked frequently in JEE Mains.

What is Order of Matrix?

The order of the matrix is referred to as the number of rows and columns that a matrix has. It indicates the dimension of a matrix and also gives the number of elements in a matrix. If a matrix has ‘m’ rows and ‘n’ columns, then the order of the matrix is said to be ‘mn’.

For example, the matrix given below has 3 rows and 4 columns. So, the order of the matrix is $3 \times 4$ and the matrix has 12 elements.

Note: In the order of a matrix, the first number always indicates the number of rows, while the second number indicates the number of columns.

How to determine the order of the Matrix

The order of a matrix is determined by the number of rows and columns it contains. For instance, if a matrix has "m" rows and "n" columns, its order is represented as "m × n."

Order of matrix = Number of row x Number of column

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 a $m \times n$ 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 $\mathrm{i}^{\text {th }}$ row and $\mathrm{j}$ th column and $\mathrm{i}=1,2, \ldots, \mathrm{m} ; \mathrm{j}=1,2, \ldots, \mathrm{n}$.
For example, to locate the entry in matrix $\mathrm{A}$ identified as $\mathrm{a}_{\mathrm{ij}}$, we look for the entry in row $\mathrm{i}$, column $\mathrm{j}$. In matrix $\mathrm{A}$, shown below, the entry in row 2, column 3 is $\mathrm{a}_{23}$.
$
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 on the basis of order

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 $1 \times 4$

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 $\mathrm{n}=1$.

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

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

Properties of order of 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.
5. Recommended Video :

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: 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 3: Find the value of $a_{32}$ in the given matrix $\left[\begin{array}{ccc}3 & 4 & -1 \\ 2 & 8 & 5 \\ 1 & 1 & -1\end{array}\right]$.
1) 1
2) -1
3) 5
4) 8

Solution:
In the given matrix
$
\left[\begin{array}{ccc}
3 & 4 & -1 \\
2 & 8 & 5 \\
1 & 1 & -1
\end{array}\right]_{3 \times 3}
$
the entry in row $\mathrm{i}$ and column $\mathrm{j}$, where $\mathrm{i}=3$ and $\mathrm{j}=2$ is
$
a_{32}=1 \text {. }
$

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: Example 5: Find the sum of all the values of $a_{i j}$ where $i$ and $j$ are the squares of any natural number in the given matrix
$
\left[\begin{array}{cccc}
3 & 5 & 6 & -1 \\
2 & 3 & 1 & 8 \\
0 & 5 & 10 & 7 \\
1 & 1 & -1 & -3
\end{array}\right]
$

1) 0
2) 6
3) 8
4) 4

Solution:

The given matrix is of order $4 \times 4$
$
\left[\begin{array}{cccc}
3 & 5 & 6 & -1 \\
2 & 3 & 1 & 8 \\
0 & 5 & 10 & 7 \\
1 & 1 & -1 & -3
\end{array}\right]
$

Therefore, the values that $i$ can take are 1 and 4 .
Similarly, the values that $j$ can take are 1 and 4.
So the required sum = $
a_{11}+a_{14}+a_{41}+a_{44}=3+(-1)+1+(-3)=0
$

Hence, the answer is the option 1.

Summary

Knowing the order of a matrix makes it simple to calculate its total number of components. In numerous mathematical and real-world applications, the order of a matrix is significant because it conveys crucial details about the matrix's composition, functions, and data representation.

Frequently asked questions

Q1) What is Matrices?

Answer: Matrices are array-like configurations of elements, numbers, or symbols.

Q2) What is meant by the order of Matrices?

Answer: The number of rows and columns of a matrix is its order. It provides the number of elements in a matrix as well as the matrix's dimension.

Q3) How to find the order of Matrices?

Answer: The order of matrices can be found by multiplying its row and column. Suppose a matrix has ‘m’ rows and ‘n’ columns. Then the order of matrices is ‘mn’

Q4) How to find the Number of rows if the order of matrices is given?

Answer: If the order of the matrix is ‘mn’ then the number of rows is m and the number of columns is n

Q5) If the Number of rows in a matrix = 3 and the number of columns = 2. Find the order of the matrix.

Answer: Order of matrix= number of row x number of column

Order of matrix = 3 x 2 =6