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# NCERT Solutions for Class 12 Maths Chapter 3 - Matrices Exercise 3.1

Last updated date: 17th Sep 2024
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## NCERT Solutions for Class 12 Maths Exercise 3.1 Chapter 3 Matrices - FREE PDF Download

Class 12 Maths Exercise 3.1 of Matrices introduces a fundamental mathematical concept essential in fields like engineering, physics, and economics. Ex 3.1 Class 12 , presented by Vedantu, focuses on matrix representation, types of Matrices, and equality. Mastering these concepts is crucial for understanding complex operations like matrix addition, subtraction, and multiplication. Familiarity with Matrices is essential for understanding further mathematical concepts that utilize them.

Table of Content
1. NCERT Solutions for Class 12 Maths Exercise 3.1 Chapter 3 Matrices - FREE PDF Download
2. Glance of NCERT Solutions for Class 12 Maths Chapter 3 Matrices - Exercise 3.1 | Vedantu
3. Topics Covered in Exercise 3.1 Class 12 Maths
4. Access NCERT Solutions for Class 12 Maths Chapter 3 – Matrices Exercise 3.1
5. Class 12 Maths Chapter 3: Exercises Breakdown
6. Other Study Materials for CBSE Class 12 Maths Chapter 3 Matrices
7. Chapter-Specific NCERT Solutions for Class 12 Maths
FAQs

## Glance of NCERT Solutions for Class 12 Maths Chapter 3 Matrices - Exercise 3.1 | Vedantu

• Class 12 Maths Exercise 3.1 focuses on Matrices, delving into their types, operations, and applications in solving mathematical problems.

• This chapter introduces the addition and subtraction of Matrices, essential for combining or comparing matrix data.

• It also covers scalar multiplication, which is used to adjust the magnitude of matrix elements.

• Matrix multiplication is another critical operation discussed, enabling the computation of resultant Matrices.

• Additionally, the chapter explains the transpose of a matrix, a process that flips a matrix over its diagonal.

• In Class 12th Maths, Chapter 3, Exercise 3.1 Matrices there are 10 Solved Questions.

## Topics Covered in Exercise 3.1 Class 12 Maths

• Introduction to Matrices

• Types of Matrices

• Scalar Multiplication

Competitive Exams after 12th Science

## Access NCERT Solutions for Class 12 Maths Chapter 3 – Matrices Exercise 3.1

1. In the matrix $A=\left[ \begin{matrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \dfrac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \\ \end{matrix} \right]$, write

i. The order of the matrix.

Ans: The order of a matrix is $m\times n$ where $m$ is the number of rows and $n$ is the number of columns. Therefore, here the order is $3\times 4$.

ii. The number of elements.

Ans: Since the order of the given matrix is $3\times 4$ therefore, the number of elements in it is $3\times 4=12$.

iii. Write the elements ${{a}_{13}},{{a}_{21}},{{a}_{33}},{{a}_{24}},{{a}_{23}}$

Ans: The elements are given as ${{a}_{mn}}$ . Therefore, here ${{a}_{13}}=19$ , ${{a}_{21}}=35$ , ${{a}_{33}}=-5$ , ${{a}_{24}}=12$ , ${{a}_{23}}=\dfrac{5}{2}$.

2. If a matrix has $24$ elements, what are the possible order it can have? What if it has $13$ elements?

Ans: The order of a matrix is $m\times n$ where $m$ is the number of rows and $n$ is the number of columns. To find the possible orders of a matrix, we have to find all the ordered pairs of natural numbers whose product is $24$ .

$\therefore \left( 1\times 24 \right),\left( 24\times 1 \right),\left( 2\times 12 \right),\left( 12\times 2 \right),\left( 3\times 8 \right),\left( 8\times 3 \right),\left( 4\times 6 \right),\left( 6\times 4 \right)$ are all the possible ordered pairs here.

If the matrix had $13$ elements, then the ordered pairs would be $\left( 1\times 13 \right)$ and $\left( 13\times 1 \right)$.

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

Ans: The order of a matrix is $m\times n$ where $m$ is the number of rows and $n$ is the number of columns. To find the possible orders of a matrix, we have to find all the ordered pairs of natural numbers whose product is $18$ .

$\therefore \left( 1\times 18 \right),\left( 18\times 1 \right),\left( 2\times 9 \right),\left( 9\times 2 \right),\left( 3\times 6 \right),\left( 6\times 3 \right)$ are all the possible ordered pairs here.

If the matrix had $5$ elements, then the ordered pairs would be $\left( 1\times 5 \right)$ and $\left( 5\times 1 \right)$.

4. Construct a $2 \times 2$matrix, $A\, = \,\left[ {{a_{ij}}} \right]$, whose elements are given by:

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

(ii) ${a_{ij}} = \frac{i}{j}$

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

Ans:

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

Elements for $2 \times 2$ matrix are: ${a_{11}},{a_{12}},{a_{21}},{a_{22}}$

${a_{11}} = \frac{{{{\left( {1 + 1} \right)}^2}}}{2}\, = \,\frac{{{{\left( 2 \right)}^2}}}{2}\, = \,2$

${a_{12}}\, = \,\frac{{{{\left( {1 + 2} \right)}^2}}}{2}\, = \,\frac{{{{\left( 3 \right)}^2}}}{2}\, = \,\frac{9}{2}$

${a_{21}}\, = \,\frac{{{{\left( {2 + 1} \right)}^2}}}{2}\, = \,\frac{{{{\left( 3 \right)}^2}}}{2}\, = \,\frac{9}{2}$

${a_{22}}\, = \,\frac{{{{\left( {2 + 2} \right)}^2}}}{2}\, = \,\frac{{{{\left( 4 \right)}^2}}}{2}\, = \,8$

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

(ii) ${a_{ij}} = \frac{i}{j}$

Elements for $2 \times 2$ matrix are: ${a_{11}},{a_{12}},{a_{21}},{a_{22}}$

${a_{11}} = \frac{1}{1}\,\, = \,1$

${a_{12}}\, = \,\frac{1}{2}\,$

${a_{21}}\, = \,\frac{2}{1}\, = \,2$

${a_{22}}\, = \,\frac{2}{2}\, = \,1$

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

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

Elements for $2 \times 2$ matrix are: ${a_{11}},{a_{12}},{a_{21}},{a_{22}}$

${a_{11}} = \frac{{{{\left( {1 + 2} \right)}^2}}}{2}\, = \,\frac{{{{\left( 3 \right)}^2}}}{2}\, = \,\frac{9}{2}$

${a_{12}}\, = \,\frac{{{{\left( {1 + 4} \right)}^2}}}{2}\, = \,\frac{{{{\left( 5 \right)}^2}}}{2}\, = \,\frac{{25}}{2}$

${a_{21}}\, = \,\frac{{{{\left( {2 + 2} \right)}^2}}}{2}\, = \,\frac{{{{\left( 4 \right)}^2}}}{2}\, = 8$

${a_{22}}\, = \,\frac{{{{\left( {2 + 4} \right)}^2}}}{2}\, = \,\frac{{{{\left( 6 \right)}^2}}}{2}\, = 1\,8$

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

5. Construct a $3\times 4$ matrix, whose elements are given by

i. ${{a}_{ij}}=\dfrac{1}{2}\left| -3i+j \right|$

Ans: Given that ${{a}_{ij}}=\dfrac{1}{2}\left| -3i+j \right|$ ,

$\therefore {{a}_{11}}=\dfrac{1}{2}\left| -3\times 1+1 \right|=1$

${{a}_{21}}=\dfrac{1}{2}\left| -3\times 2+1 \right|=\dfrac{5}{2}$

${{a}_{31}}=\dfrac{1}{2}\left| -3\times 3+1 \right|=4$

${{a}_{12}}=\dfrac{1}{2}\left| -3\times 1+2 \right|=\dfrac{1}{2}$

${{a}_{22}}=\dfrac{1}{2}\left| -3\times 2+2 \right|=2$

${{a}_{32}}=\dfrac{1}{2}\left| -3\times 3+2 \right|=\dfrac{7}{2}$

${{a}_{13}}=\dfrac{1}{2}\left| -3\times 1+3 \right|=0$

${{a}_{23}}=\dfrac{1}{2}\left| -3\times 2+3 \right|=\dfrac{3}{2}$

${{a}_{33}}=\dfrac{1}{2}\left| -3\times 3+3 \right|=3$

${{a}_{14}}=\dfrac{1}{2}\left| -3\times 1+4 \right|=\dfrac{1}{2}$

${{a}_{24}}=\dfrac{1}{2}\left| -3\times 2+4 \right|=1$

${{a}_{34}}=\dfrac{1}{2}\left| -3\times 3+4 \right|=\dfrac{5}{2}$

Thus, the required matrix is $A=\left[ \begin{matrix} 1 & \dfrac{1}{2} & 0 & \dfrac{1}{2} \\ \dfrac{5}{2} & 2 & \dfrac{3}{2} & 1 \\ 4 & \dfrac{7}{2} & 3 & \dfrac{5}{2} \\ \end{matrix} \right]$.

ii. ${{a}_{ij}}=2i-j$

Ans: A $3\times 4$ matrix is given by $A=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} & {{a}_{14}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} & {{a}_{24}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} & {{a}_{34}} \\ \end{matrix} \right]$

Given that ${{a}_{ij}}=2i-j$ ,

$\therefore {{a}_{11}}=2\times 1-1=1$

${{a}_{21}}=2\times 2-1=3$

${{a}_{31}}=2\times 3-1=5$

${{a}_{12}}=2\times 1-2=0$

${{a}_{22}}=2\times 2-2=4$

${{a}_{32}}=2\times 3-2=4$

${{a}_{13}}=2\times 1-3=-1$

${{a}_{23}}=2\times 2-3=1$

${{a}_{33}}=2\times 3-3=3$

${{a}_{14}}=2\times 1-4=-2$

${{a}_{24}}=2\times 2-4=0$

${{a}_{34}}=2\times 3-4=2$

Thus, the required matrix is $A=\left[ \begin{matrix} 1 & 0 & -1 & -2 \\ 3 & 2 & 1 & 0 \\ 5 & 4 & 3 & 2 \\ \end{matrix} \right]$.

6. Find the value of $x,y,z$ from the following equation:

i. $\left[ \begin{matrix} 4 & 3 \\ x & 5 \\ \end{matrix} \right]=\left[ \begin{matrix} y & z \\ 1 & 5 \\ \end{matrix} \right]$

Ans: Given $\left[ \begin{matrix} 4 & 3 \\ x & 5 \\ \end{matrix} \right]=\left[ \begin{matrix} y & z \\ 1 & 5 \\ \end{matrix} \right]$

Comparing the corresponding elements we get,

$x=1,y=4,z=3$

ii. $\left[ \begin{matrix} x+y & 2 \\ 5+z & xy \\ \end{matrix} \right]=\left[ \begin{matrix} 6 & 2 \\ 5 & 8 \\ \end{matrix} \right]$

Ans: Given $\left[ \begin{matrix} x+y & 2 \\ 5+z & xy \\ \end{matrix} \right]=\left[ \begin{matrix} 6 & 2 \\ 5 & 8 \\ \end{matrix} \right]$

Comparing the corresponding elements we get,

$x+y=6,xy=8,5+z=5$

Now, $\because 5+z=5$

$\Rightarrow z=0$

We know that, ${{\left( x-y \right)}^{2}}={{\left( x+y \right)}^{2}}-4xy$

$\Rightarrow {{\left( x-y \right)}^{2}}=36-32$

$\Rightarrow \left( x-y \right)=\pm 2$

When $\left( x-y \right)=2$ and $\left( x+y \right)=6$,

We get $x=4,y=2$

When $\left( x-y \right)=-2$ and $\left( x+y \right)=6$,

We get $x=2,y=4$

$\therefore x=4,y=2,z=0$ or $\therefore x=2,y=4,z=0$

iii. $\left[ \begin{matrix} x+y+z \\ x+z \\ y+z \\ \end{matrix} \right]=\left[ \begin{matrix} 9 \\ 5 \\ 7 \\ \end{matrix} \right]$

Ans: Given $\left[ \begin{matrix} x+y+z \\ x+z \\ y+z \\ \end{matrix} \right]=\left[ \begin{matrix} 9 \\ 5 \\ 7 \\ \end{matrix} \right]$

Comparing the corresponding elements we get,

$x+y+z=9$                      …(1)

$x+z=5$                              …(2)

$y+z=7$                              …(3)

From equation (1) and (2),

$y+5=9$

$\Rightarrow y=4$

From equation (3) we have,

$4+z=7$

$\Rightarrow z=3$

$x+z=5$

$\Rightarrow x=2$

$\therefore x=2,y=4,z=3$

7. Find the value of $a,b,c,d$ from the equation:

$\left[ \begin{matrix} a-b & 2a+c \\ 2a-b & 3c+d \\ \end{matrix} \right]=\left[ \begin{matrix} -1 & 5 \\ 0 & 13 \\ \end{matrix} \right]$

Ans: Given $\left[ \begin{matrix} a-b & 2a+c \\ 2a-b & 3c+d \\ \end{matrix} \right]=\left[ \begin{matrix} -1 & 5 \\ 0 & 13 \\ \end{matrix} \right]$

Comparing the corresponding elements we get,

$a-b=-1$                              …(1)

$2a-b=0$                              …(2)

$2a+c=5$                              …(3)

$3c+d=13$                                        …(4)

From equation (2),

$b=2a$

From equation (1),

$a-2a=-1$

$\Rightarrow a=1$

$\Rightarrow b=2$

From equation (3),

$2\times 1+c=5$

$\Rightarrow c=3$

From equation (4),

$3\times 3+d=13$

$\Rightarrow d=4$

$\therefore a=1,b=2,c=3,d=4$

8. $A={{\left[ {{a}_{y}} \right]}_{m\times n}}$ is a square matrix, if

1. $m<n$

2. $m>n$

3. $m=n$

4. None of these

Ans: A given matrix is said to be a square matrix if the number of rows is equal to the number of columns.

$\therefore A={{\left[ {{a}_{y}} \right]}_{m\times n}}$ is a square matrix if, $m=n$.

Thus, option (C) is correct.

9. Which of the given values of $x$ and $y$ make the following pair of matrices equal $\left[ \begin{matrix} 3x+7 & 5 \\ y+1 & 2-3x \\ \end{matrix} \right]=\left[ \begin{matrix} 0 & y-2 \\ 8 & 4 \\ \end{matrix} \right]$

1. $x=\dfrac{-1}{3},y=7$

2. Not possible to find

3. $y=7,x=\dfrac{-2}{3}$

4. $x=\dfrac{-1}{3},y=\dfrac{-2}{3}$

Ans: Given $\left[ \begin{matrix} 3x+7 & 5 \\ y+1 & 2-3x \\ \end{matrix} \right]=\left[ \begin{matrix} 0 & y-2 \\ 8 & 4 \\ \end{matrix} \right]$

Comparing the corresponding elements we get,

$3x+7=0$

$\Rightarrow x=-\dfrac{7}{3}$

$y-2=5$

$\Rightarrow y=7$

$y+1=8$

$\Rightarrow y=7$

$2-3x=4$

$\Rightarrow x=-\dfrac{2}{3}$

Since we get two different values of $x$ ,which is not possible. It is not possible to find the values of $x$ and $y$ for which the given matrices are equal.

Thus, the correct option is (B).

10. The number of all possible matrices of order $3\times 3$ with each entry $0$ or $1$ is:

1. $27$

2. $18$

3. $81$

4. $512$

Ans: Given a matrix of the order $3\times 3$ has nine elements and each of these elements can be either $0$ or $1$ .

Now, each of the nine elements can be filled in two possible ways.

Therefore, the required number of possible matrices is ${{2}^{9}}=512$.

## Conclusion

Class 12 Maths Exercise 3.1 on Matrices, is fundamental for understanding the basics of matrices, which are crucial for advanced mathematics. This exercise focuses on the definition, types, and operations of matrices. It's important to grasp concepts like the order of a matrix, equality of matrices, and different types of matrices such as row, column, and square matrices. Pay special attention to solving problems related to these topics, as a strong foundation in these basics will help in understanding more complex matrix operations in later exercises.

## Class 12 Maths Chapter 3: Exercises Breakdown

 Exercise Number of Questions Exercise 3.2 22 Questions & Solutions (3 Short Answers, 19 Long Answers) Exercise 3.3 12 Questions & Solutions (4 Short Answers, 8 Long Answers) Exercise 3.4 1 Question & Solution Miscellaneous Exercise 11 Questions & Solutions

## Other Study Materials for CBSE Class 12 Maths Chapter 3 Matrices

 S.No. Important Links for Chapter 3 Matrices 1 Class 12 Matrices Important Questions 2 Class 12 Matrices Revision Notes 3 Class 12 Matrices Important Formulas 4 Class 12 Matrices NCERT Exemplar Solution 5 Class 12 Matrices RS Aggarwal Solutions

## Chapter-Specific NCERT Solutions for Class 12 Maths

Given below are the chapter-wise NCERT Solutions for Class 12 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

## FAQs on NCERT Solutions for Class 12 Maths Chapter 3 - Matrices Exercise 3.1

1. Does Vedantu offer exercise-wise NCERT Solutions for class 12 maths chapter 3 exercise 3.1?

Yes, Vedantu offers exercise-wise NCERT Solutions for Class 12 Maths Chapter 3 Matrices. Vedantu is the leading ed-tech company where students can avail NCERT Solutions for all subjects. NCERT Solutions for Class 12 Maths Chapter 3 Matrices Exercise 3.1 is available on the platform for free. All you need to do is download the PDF file. These solutions are provided by expert Maths tutors at Vedantu. Students can clear doubts related to the particular exercise or other exercises of Chapter 3 Matrices by visiting Vedantu’s site.

2. What are the benefits of NCERT Solutions for Class 12 Maths Chapter 3 Matrices for Ex 3.1?

Students must refer to online exercise-wise NCERT Solutions for Class 12 Maths Chapter 3 Matrices to understand the chapter in a better manner. These solutions, prepared by online learning sites like Vedantu, are the most comprehensive study material. It provides instant doubt resolution and students can finish off the exercise easily. The solutions are available in the downloadable PDF format on Vedantu. This allows students to study at the comfort of their homes. NCERT Solutions for Class 12 Mathematics Chapter 3 Matrices Ex 3.1 and other exercises include step-by-step explanations of each problem.

3. How to define a Matrix? What is the Order of a Matrix?

A matrix can be defined as a rectangular array of elements which can be numbers, symbols or expressions. These elements are generally arranged in rows and columns. The order of a matrix is nothing but the number of rows and columns. The plural of matrix is matrices.

4. Name the different Types of Matrices.

The different types of matrices are as follows:

• Row Matrix

• Column Matrix

• Null Matrix

• Diagonal Matrix

• Square Matrix

• Upper Triangular Matrix

• Lower Triangular Matrix

• Symmetric Matrix

• Anti-symmetric Matrix

Students can learn about each type of matrix in detail by studying Class 12 Maths Chapter 3. In case of any doubts, students can download the solutions available on Vedantu and even enrol for LIVE Classes on the platform.

5. What is the use of Matrices according to Chapter 3 of Class 12 Maths?

Chapter 3 of Class 12 Maths is regarding Matrices. This is a type of numerical where you will get a few numbers or elements arranged inside the third brackets. You will be able to perform addition, subtraction, multiplication, etc. on the matrices. The use of learning matrices is their application in the area of vectors. When you study vectors, you will have to transform the linear equations to and from the vector field. So this becomes important to study and understand this chapter well.

6. How do you determine the number of elements from a matrix?

The Chapter 3 Matrices of Class 12 will introduce you to a few new terms among which elements are one. Elements are the numbers arranged in rows and columns inside a matrix bracket. The number of elements can be very easily determined or calculated by simply multiplying the number of rows with the number of columns present inside the third bracket. This will give the number of total elements inside the matrix. For example, if there are three rows and four columns then the number of elements will be 3*4= 12.

7. How can the order of a matrix be calculated?

Order of the matrix is basically the representation of the matrix in terms of rows and columns. If there are m number of rows in a matrix and n number of columns, then the order of the matrix will be written as m*n. The order of the matrix is important since it helps us to calculate the number of elements and also to understand the nature of the matrix. The order of matrices needs to be similar in order to be able to equate two or more numbers of matrices. If the order is not the same then it cannot be equated.

8. What happens to the elements of two different matrices when the matrices are equal?

When you have two equal matrices, with an equal sign given in between them, then we will know that the corresponding elements of those two matrices are equal as well. For example, if the first element of the first matrix is two and the first element of the second matrix is unknown that is represented as ‘x’ then the value of x will be equal to two. This is how we can find the unknown elements from equal matrices.

9. Is the Matrix Chapter important for boards?

Yes, the Matrix Chapter from Class 12 is very important. This is because every year there are guaranteed questions from this chapter that come in your board exam. So it becomes necessary to have clear concepts in this Chapter and also be perfect with the sums from this Chapter. If you practice all the NCERT questions and solutions then it will become easier for you to score well in the Maths Class 12 exam. Also, if you are willing to appear for competitive exams like JEE Mains etc after your boards then this chapter must be studied well.

10. What topics are covered in NCERT Ex 3.1 Class 12 Maths?

Ex 3.1 Class 12 focuses on the basics of Matrices, types of Matrices (like row matrix, column matrix, square matrix, diagonal, scalar, and identity Matrices), and matrix operations such as addition and scalar multiplication.

11. What should I focus on in ex 3.1 class 12 for exam preparation?

In Class 12 Maths Chapter 3 Exercise 3.1 focuses on understanding how to correctly perform matrix operations and identifying different types of Matrices. These are crucial for both theoretical and practical questions. Additionally, make sure you can easily distinguish between different types of Matrices like row, column, and square Matrices. Practice solving problems related to matrix addition and scalar multiplication to build your confidence and accuracy.

12. What should I focus on in 12th Maths Exercise 3.1 for exam preparation?

In class 12 ex 3.1 focus on understanding how to correctly perform matrix operations and identifying different types of matrices. Practice problems involving matrix addition and scalar multiplication. Make sure you can easily distinguish between types like row, column, and square matrices. Regular practice will help you build confidence and accuracy.

13. Are the questions in Class 12 Ex 3.1 complex?

The questions in 12th Maths Exercise 3.1 are generally simple and straightforward. They mainly involve basic matrix operations and identifying different types of matrices. With some practice, you should find them easy to solve. These questions are designed to help you understand the fundamentals of matrices.