CBSE Class 12 Mathematics Chapter 3: Matrices NCERT Solutions

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 12 PDF

This chapter introduces students to the fundamental concepts of matrices, a crucial topic in Class 12 Mathematics. The NCERT Solutions for Chapter 3 cover the definition of matrices, their order, the number of elements, and how to identify specific elements. It also delves into constructing matrices based on given element formulas. These solutions provide step-by-step guidance for solving problems related to matrix order, element identification, and matrix construction, ensuring a clear understanding of the basic principles. This resource is designed to aid students in mastering these foundational concepts, preparing them effectively for examinations by reinforcing their knowledge through clear explanations and solved examples.

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Board	CBSE
Class	Class 12
Subject	Mathematics
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 3

Chapter summary

Chapter 3 of the NCERT Class 12 Mathematics textbook focuses on Matrices. These solutions cover the basics, including defining a matrix, determining its order (number of rows and columns), and calculating the total number of elements. The exercises also involve identifying specific elements within a matrix using their row and column indices and constructing matrices of a given order based on provided formulas for their elements. This chapter lays the groundwork for more advanced matrix operations studied later.

Learning outcomes

Understand the definition and notation of a matrix.
Determine the order of a given matrix.
Calculate the total number of elements in a matrix.
Identify specific elements of a matrix using their indices (a_ij).
Construct a matrix of a specified order based on given element rules.
Differentiate between matrices with different numbers of elements and their possible orders.

Topics covered

Paper topics

Introduction to Matrices
Order of a Matrix
Number of Elements in a Matrix
Identifying Matrix Elements
Construction of Matrices
Matrix Notation (a_ij)
Row and Column Concepts
Basic Matrix Properties

Important topics

Order of a Matrix
Identifying Elements (a_ij)
Constructing Matrices
Number of Elements

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Questions and Solutions

Question 1

Given the matrix

A = \begin{pmatrix}
    2 & 5 & 19 & -7 \\ 
    35 & -2 & \frac{5}{2} & 12 \\ 
    \sqrt{3} & 1 & -5 & 17
  \end{pmatrix}

Write:

The order of the matrix
The number of elements
The elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$

Solution:

The given matrix is:

A = \begin{pmatrix}
    2 & 5 & 19 & -7 \\ 
    35 & -2 & \frac{5}{2} & 12 \\ 
    \sqrt{3} & 1 & -5 & 17
  \end{pmatrix}

To determine the order of the matrix, we count the number of rows and columns.

Number of rows (m) = 3

Number of columns (n) = 4

The order of the matrix is given by m x n, which is $3 \times 4$ .
The number of elements in the matrix is the product of the number of rows and columns, i.e., $m \times n = 3 \times 4 = 12$ .
The elements are identified by their position (row index i, column index j) as a_{ij}:
- $a_{13}$ is the element in the 1st row and 3rd column, which is 19.
- $a_{21}$ is the element in the 2nd row and 1st column, which is 35.
- $a_{33}$ is the element in the 3rd row and 3rd column, which is -5.
- $a_{24}$ is the element in the 2nd row and 4th column, which is 12.
- $a_{23}$ is the element in the 2nd row and 3rd column, which is $\frac{5}{2}$ .

Question 2

If a matrix has 24 elements, what are the possible orders it can have? What if it has 13 elements?

Solution:

The number of elements in a matrix is the product of its order (number of rows m and number of columns n), i.e., m \times n.

To find the possible orders for a matrix with a given number of elements, we need to find all pairs of natural numbers whose product equals that number.

Case 1: Matrix with 24 elements

We need to find pairs of natural numbers (m, n) such that m \times n = 24. The possible pairs are:

$(1, 24)$
$(2, 12)$
$(3, 8)$
$(4, 6)$
$(6, 4)$
$(8, 3)$
$(12, 2)$
$(24, 1)$

Therefore, the possible orders for a matrix with 24 elements are $1 \times 24$ , $2 \times 12$ , $3 \times 8$ , $4 \times 6$ , $6 \times 4$ , $8 \times 3$ , $12 \times 2$ , and $24 \times 1$ .

Case 2: Matrix with 13 elements

We need to find pairs of natural numbers $(m, n)$ such that $m \times n = 13$ . Since 13 is a prime number, its only natural number factors are 1 and 13.

$(1, 13)$
$(13, 1)$

Therefore, the possible orders for a matrix with 13 elements are $1 \times 13$ and $13 \times 1$ .

Question 3

If a matrix has 18 elements, what are the possible orders it can have? What if it has 5 elements?

Solution:

Similar to the previous question, the number of elements in a matrix is the product of its order ( $m \times n$ ). We find all pairs of natural numbers whose product equals the given number of elements.

Case 1: Matrix with 18 elements

We look for pairs $(m, n)$ such that $m \times n = 18$ . The possible pairs are:

$(1, 18)$
$(2, 9)$
$(3, 6)$
$(6, 3)$
$(9, 2)$
$(18, 1)$

Thus, the possible orders for a matrix with 18 elements are $1 \times 18$ , $2 \times 9$ , $3 \times 6$ , $6 \times 3$ , $9 \times 2$ , and $18 \times 1$ .

Case 2: Matrix with 5 elements

We look for pairs $(m, n)$ such that $m \times n = 5$ . Since 5 is a prime number, its only natural number factors are 1 and 5.

$(1, 5)$
$(5, 1)$

Therefore, the possible orders for a matrix with 5 elements are $1 \times 5$ and $5 \times 1$ .

Question 4

Construct a

2 \times 2

matrix,

A = [a_{ij}]

, whose elements are given by:

$a_{ij} = \frac{(i+j)^2}{2}$
$a_{ij} = \frac{i}{j}$
$a_{ij} = \frac{(i+2j)^2}{2}$

Solution:

A general $2 \times 2$ matrix has the form:

A = \begin{pmatrix}
    a_{11} & a_{12} \\ 
    a_{21} & a_{22}
  \end{pmatrix}

We will construct the matrix for each given condition by substituting the values of $i$ (row number) and $j$ (column number) from 1 to 2.

Given $a_{ij} = \frac{(i+j)^2}{2}$

Calculate each element:

$a_{11} = \frac{(1+1)^2}{2} = \frac{2^2}{2} = \frac{4}{2} = 2$
$a_{12} = \frac{(1+2)^2}{2} = \frac{3^2}{2} = \frac{9}{2}$
$a_{21} = \frac{(2+1)^2}{2} = \frac{3^2}{2} = \frac{9}{2}$
$a_{22} = \frac{(2+2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8$

So, the matrix is: $A = \begin{pmatrix}
2 & \frac{9}{2} \\
\frac{9}{2} & 8
\end{pmatrix}$

Given $a_{ij} = \frac{i}{j}$

Calculate each element:

$a_{11} = \frac{1}{1} = 1$
$a_{12} = \frac{1}{2}$
$a_{21} = \frac{2}{1} = 2$
$a_{22} = \frac{2}{2} = 1$

So, the matrix is: $A = \begin{pmatrix}
1 & \frac{1}{2} \\
2 & 1
\end{pmatrix}$

Given $a_{ij} = \frac{(i+2j)^2}{2}$

Calculate each element:

$a_{11} = \frac{(1+2(1))^2}{2} = \frac{(1+2)^2}{2} = \frac{3^2}{2} = \frac{9}{2}$
$a_{12} = \frac{(1+2(2))^2}{2} = \frac{(1+4)^2}{2} = \frac{5^2}{2} = \frac{25}{2}$
$a_{21} = \frac{(2+2(1))^2}{2} = \frac{(2+2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8$
$a_{22} = \frac{(2+2(2))^2}{2} = \frac{(2+4)^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18$

So, the matrix is: $A = \begin{pmatrix}
\frac{9}{2} & \frac{25}{2} \\
8 & 18
\end{pmatrix}$

Common mistakes

Confusing the number of rows with the number of columns when stating the order.
Incorrectly identifying elements based on their row and column indices.
Errors in calculating the total number of elements from the order.
Mistakes in substituting 'i' and 'j' values when constructing matrices from formulas.

Revision tips

Clearly understand the notation for matrix order (m x n) and elements (a_ij).
Practice identifying elements by carefully noting their row and column positions.
When constructing matrices, systematically calculate each element using the given formula.
For problems asking for possible orders, find all factor pairs of the total number of elements.

Practice MCQs

Q1. What is the order of a matrix with 5 rows and 3 columns?

Q2. If a matrix has 12 elements, which of the following is NOT a possible order?

Q3. In a matrix $A =$ [a_ij], what does the subscript '13' in $a_13$ represent?

Q4. How many elements are there in a matrix of order 3 x 3?

Frequently asked questions

What is the main focus of Chapter 3: Matrices in Class 12 Maths?

Chapter 3 introduces the fundamental concepts of matrices, including their definition, order, the number of elements, and how to identify specific elements using notation like a_ij. It also covers the construction of matrices based on given rules.

How do I find the order of a matrix?

The order of a matrix is determined by the number of rows and the number of columns it has. It is expressed as 'm x n', where 'm' is the number of rows and 'n' is the number of columns.

What does a_ij mean in a matrix?

In a matrix A, the notation a_ij represents the element located in the i-th row and the j-th column of the matrix.

How can I determine the possible orders for a matrix with a given number of elements?

To find the possible orders, you need to find all pairs of natural numbers (factors) whose product equals the total number of elements. For example, a matrix with 12 elements can have orders like 1x12, 12x1, 2x6, 6x2, 3x4, and 4x3.

Are these NCERT Solutions helpful for exam preparation?

Yes, these solutions provide clear, step-by-step explanations for all exercises in Chapter 3, helping students understand the concepts thoroughly and practice problem-solving for their exams.

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CBSE Class 12 Mathematics Chapter 3: Matrices NCERT Solutions

Quick info

Chapter summary

Learning outcomes

Topics covered

Paper topics

Important topics

PDF preview

Questions and Solutions

Question 1

Question 2

Question 3

Question 4

Common mistakes

Revision tips

Practice MCQs

Frequently asked questions

Important links