NCERT Solutions for Exercise 3.1 Class 12 Maths Chapter 3

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

Edited By Ramraj Saini | Updated on Dec 03, 2023 02:58 PM IST | #CBSE Class 12th

NCERT Solutions For Class 12 Maths Chapter 3 Exercise 3.1

NCERT Solutions for Exercise 3.1 Class 12 Maths Chapter 3 Matrices are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. Matrix is an ordered rectangular array of numbers or functions called the elements or the entries of the matrix. It is a very important tool used in genetics, science, sociology, modern psychology, economics, and industrial management. In NCERT solutions for class 12 maths matrices exercise 3.1, you will get questions solved in a step-by-step manner. It covers the matrix, the order of the matrix, and different types of matrix. First, try to solve NCERT textbook problems on your own. If you are finding difficulties in solving them, you can take help from exercise 3.1 chapter 3 maths solutions. The important topics like Square matrix, Row matrix, Column matrix, Diagonal matrix, Scalar matrix, Zero matrix, Identity matrix, and Equality of matrices are also covered in the exercise 3.1 class 12 maths.

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12th class Maths exercise 3.1 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

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Assess NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.1

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Matrices Exercise 3.1

Question:1(i).In the matrix $A = \begin{bmatrix} 2& 5 &19 &-7 \\ 35 & -2 & \frac{5}{2} &12 \\ \sqrt3& 1 &-5 &17 \end{bmatrix}$ , write:

The order of the matrix

Answer:

$A = \begin{bmatrix} 2& 5 &19 &-7 \\ 35 & -2 & \frac{5}{2} &12 \\ \sqrt3& 1 &-5 &17 \end{bmatrix}$

(i) The order of the matrix = number of row $\times$ number of columns $= 3\times 4$ .

Question 1(ii). In the matrix $A = \begin{bmatrix}2&5&19&-7&\\ 35& -2&\frac{5}{2}&12\\\sqrt3&1&-5&17 \end{bmatrix}$ , write:

The number of elements

Answer:

$A = \begin{bmatrix}2&5&19&-7&\\ 35& -2&\frac{5}{2}&12\\\sqrt3&1&-5&17 \end{bmatrix}$

(ii) The number of elements $3\times 4=12$ .

Question 1(iii). In the matrix $A = \begin{bmatrix}2&5&19&-7&\\35&-2&\frac{5}{2}&12\\\sqrt3&1&-5&17 \end{bmatrix}$ , write:

Write the elements a₁₃, a₂₁, a₃₃, a₂₄, a₂₃

Answer:

$A = \begin{bmatrix}2&5&19&-7&\\35&-2&\frac{5}{2}&12\\\sqrt3&1&-5&17 \end{bmatrix}$

(iii) An element $a_{ij}$ implies the element in raw number i and column number j.

$a_1_3= 19$ $a_2_1= 35$

$a_3_3= -5$ $a_2_4= 12$

$a_2_3= \frac{5}{2}$

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

Answer:

A matrix has 24 elements.

The possible orders are :

$1\times 24,24\times 1,2\times 12,12\times 2,3\times 8,8\times 3,4\times 6 \, \, and\, \, 6\times 4$ .

If it has 13 elements, then possible orders 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?

Answer:

A matrix has 18 elements.

The possible orders are as given below

$1\times 18,18\times 1,2\times 9,9\times 2,3\times 6\, \, \, and\, \, \, \, 6\times 3$

If it has 5 elements, then possible orders are :

$1\times 5\, \, \, and \, \, \, \, 5\times 1$ .

Question 4(i). Construct a 2 × 2 matrix, $A = [a_{ij} ]$ whose elements are given by:

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

Answer:

$A = [a_{ij} ]$

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

Each element of this matrix is calculated as follows

$a_1_1 = \frac{(1+1)^{2}}{2} =\frac{2^{2}}{2}=\frac{4}{2}=2$ $a_2_2 = \frac{(2+2)^{2}}{2} =\frac{4^{2}}{2}=\frac{16}{2}=8$

$a_1_2 = \frac{(1+2)^{2}}{2} =\frac{3^{2}}{2}=\frac{9}{2}=4.5$ $a_2_1 = \frac{(2+1)^{2}}{2} =\frac{3^{2}}{2}=\frac{9}{2}=4.5$

Matrix A is given by

$A = \begin{bmatrix} 2&4.5 \\4.5 & 8 \end{bmatrix}$

Question 4(ii). Construct a 2 × 2 matrix, $A = [a_{ij} ]$ , whose elements are given by:

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

Answer:

A 2 × 2 matrix, $A = [a_{ij} ]$

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

$a_1_1 = \frac{1}{1}=1$ $a_2_2 = \frac{2}{2}=1$

$a_1_2 = \frac{1}{2}$ $a_2_1 = \frac{2}{1}=2$

Hence, the matrix is

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

Question 4(iii). Construct a 2 × 2 matrix, $A = [a_{ij} ]$ , whose elements are given by:

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

Answer:

(iii)

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

$a_1_1 = \frac{(1+(2\times 1))^{2}}{2}= \frac{(1+2)^{2}}{2}=\frac{3^{2}}{2}=\frac{9}{2}$ $a_2_2 = \frac{(2+(2\times 2))^{2}}{2}= \frac{(2+4)^{2}}{2}=\frac{6^{2}}{2}=\frac{36}{2}=18$

$a_2_1 = \frac{(2+(2\times 1))^{2}}{2}= \frac{(2+2)^{2}}{2}=\frac{4^{2}}{2}=\frac{16}{2}=8$ $a_1_2 = \frac{(1+(2\times 2))^{2}}{2}= \frac{(1+4)^{2}}{2}=\frac{5^{2}}{2}=\frac{25}{2}$

Hence, the matrix is given by

$A = \begin{bmatrix} \frac{9}{2}& \frac{25}{2} \\ 8 & 18 \end{bmatrix}$

Question 5(i). Construct a 3 × 4 matrix, whose elements are given by:

$a_{ij} = \frac{1}{2}|-3i + j|$

Answer:

(i)

$a_{ij} = \frac{1}{2}|-3i + j|$

$a_1_1 = \frac{\left | -3+1 \right |}{2}=\frac{2}{2}=1$ $a_1_2 = \frac{\left | (-3\times 1)+2 \right |}{2}=\frac{1}{2}$ $a_1_3 = \frac{\left | (-3\times 1)+3 \right |}{2}=0$

$a_2_1 = \frac{\left | (-3\times 2)+1 \right |}{2}=\frac{5}{2}$ $a_2_2 = \frac{\left | (-3\times 2)+2 \right |}{2}=\frac{4}{2}=2$ $a_2_3 = \frac{\left | (-3\times 2)+3 \right |}{2}=\frac{\left | -6+3 \right |}{2}=\frac{\left | -3 \right |}{2} =\frac{3}{2}$

$a_3_1 = \frac{\left | (-3\times 3)+1 \right |}{2}=\frac{8}{2}=4$ $a_3_2 = \frac{\left | (-3\times 3)+2 \right |}{2}=\frac{7}{2}$ $a_3_3 = \frac{\left | (-3\times 3)+3 \right |}{2}=\frac{\left | -9+3 \right |}{2}=\frac{\left | -6 \right |}{2} =\frac{6}{2}=3$

$a_1_4 = \frac{\left | (-3\times 1)+4 \right |}{2}=\frac{\left | -3+4 \right |}{2}=\frac{\left | 1 \right |}{2} =\frac{1}{2}$ $a_2_4 = \frac{\left | (-3\times 2)+4 \right |}{2}=\frac{\left | -6+4 \right |}{2}=\frac{\left | -2 \right |}{2} =\frac{2}{2}=1$

$a_3_4 = \frac{\left | (-3\times 3)+4 \right |}{2}=\frac{\left | -9+4 \right |}{2}=\frac{\left | -5 \right |}{2} =\frac{5}{2}$

Hence, the required matrix of the given order is

$A = \begin{bmatrix} 1& \frac{1}{2} & 0&\frac{1}{2} \\ \frac{5}{2} & 2&\frac{3}{2}&1 \\4&\frac{7}{2}&3&\frac{5}{2}\end{bmatrix}$

Question 5(ii) Construct a 3 × 4 matrix, whose elements are given by:

$a_{ij} = 2i - j$

Answer:

A 3 × 4 matrix,

(ii) $a_{ij} = 2i - j$

$a_1_1 = 2\times 1-1 =2-1=1$ $a_1_2 = 2\times 1-2 =2-2=0$ $a_1_3 = 2\times 1-3 =2-3=-1$

$a_2_1 = 2\times 2-1 =4-1=3$ $a_2_2= 2\times 2-2 =4-2=2$ $a_2_3 = 2\times 2-3 =4-3=1$ $a_3_1 = 2\times 3-1 =6-1=5$ $a_3_2 = 2\times 3-2 =6-2=4$ $a_3_3 = 2\times 3-3 =6-3=3$

$a_1_4 = 2\times 1-4 =2-4=-2$ $a_2_4= 2\times 2-4 =4-4=0$ $a_3_4= 2\times 3-4 =6-4=2$

Hence, the matrix is

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

Question 6(i). Find the values of x, y and z from the following equations:

$\begin{bmatrix}4&3\\x&5 \end{bmatrix} = \begin{bmatrix}y&z\\1&5 \end{bmatrix}$

Answer:

(i) $\begin{bmatrix}4&3\\x&5 \end{bmatrix} = \begin{bmatrix}y&z\\1&5 \end{bmatrix}$

If two matrices are equal, then their corresponding elements are also equal.

$\therefore$ $x=1\, \, \, ,\, \, \, y=4\, \, \, \, and\, \, \, \, z=3$

Question 6(ii) Find the values of x, y and z from the following equations:

$\begin{bmatrix} x +y & 2\\ 5 + z & xy \end{bmatrix} = \begin{bmatrix} 6 &2 \\ 5 & 8 \end{bmatrix}$

Answer:

(ii)

$\begin{bmatrix} x +y & 2\\ 5 + z & xy \end{bmatrix} = \begin{bmatrix} 6 &2 \\ 5 & 8 \end{bmatrix}$

If two matrices are equal, then their corresponding elements are also equal.

$\therefore$ $x+y=6$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot (i)$

$x=6-y$

$xy=8$ $\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot (ii)$

Solving equation (i) and (ii) ,

$(6-y)y =8$

$6y-y^{2}=8$

$y^{2}-6y+8=0$

solving this equation we get,

$y=4 \, \, and\, \, y=2$

Putting the values of y, we get

$x=2 \, \, and\, \, x=4$

And also equating the first element of the second raw

$5+z = 5$ , $z=0$

Hence,

$x=2,y=4,z=0\, \, \, \, \, and\, \, \, \, \, \, x=4,y=2,z=0$

Question 6(iii) Find the values of x, y and z from the following equations

$\begin{bmatrix} x + y + z\\ x + z \\ y + z \end{bmatrix} = \begin{bmatrix} 9\\5 \\7 \end{bmatrix}$

Answer:

(iii)

$\begin{bmatrix} x + y + z\\ x + z \\ y + z \end{bmatrix} = \begin{bmatrix} 9\\5 \\7 \end{bmatrix}$

If two matrices are equal, then their corresponding elements are also equal

$x+y+z=9........(1)$

$x+z=5..............(2)$

$y+z=7..............(3)$

subtracting (2) from (1) we will get y=4

substituting the value of y in equation (3) we will get z=3

now substituting the value of z in equation (2) we will get x=2

therefore,

$x=2$ , $y=4$ and $z=3$

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

$\begin{bmatrix} a -b & 2a + c\\ 2a - b & 3c + d \end{bmatrix} = \begin{bmatrix} -1 & 5\\ 0 & 13 \end{bmatrix}$

Answer:

$\begin{bmatrix} a -b & 2a + c\\ 2a - b & 3c + d \end{bmatrix} = \begin{bmatrix} -1 & 5\\ 0 & 13 \end{bmatrix}$

If two matrices are equal, then their corresponding elements are also equal

$a-b=-1$ $.............................1$

$2a+c=5$ $.............................2$

$2a-b=0$ $.............................3$

$3c+d=13$ $.............................4$

Solving equation 1 and 3 , we get

$a=1 \, \, \, \, and \, \, \, \, b=2$

Putting the value of a in equation 2, we get

$c=3$

Putting the value of c in equation 4 , we get

$d=4$

Question 8. $A = [a_{ij}]_{m\times n}$ is a square matrix, if

(A) $m <n$

(B) $m >n$

(D) None of these

Answer:

A square matrix has the number of rows and columns equal.

Thus, for $A = [a_{ij}]_{m\times n}$ to be a square matrix m and n should be equal.

$\therefore m=n$

Option (c) is correct.

Question 9. Which of the given values of x and y make the following pair of matrices equal

$\begin{bmatrix} 3x + 7 &5 \\ y + 1 & 2 -3x \end{bmatrix}$ , $\begin{bmatrix} 0 & y - 2 \\ 8 & 4 \end{bmatrix}$

(A) $x = \frac{-1}{3}, y = 7$

(B) Not possible to find

(D) $x = \frac{-1}{3}, y = \frac{-2}{3}$

Answer:

Given, $\begin{bmatrix} 3x + 7 &5 \\ y + 1 & 2 -3x \end{bmatrix}$ $=\begin{bmatrix} 0 & y - 2 \\ 8 & 4 \end{bmatrix}$

If two matrices are equal, then their corresponding elements are also equal

$3x+7=0\Rightarrow x=\frac{-7}{3}$

$y-2=5 \Rightarrow y=5+2=7$

$y+1=8\Rightarrow y=8-1=7$

$2-3x=4\Rightarrow 3x=2-4\Rightarrow 3x=-2\Rightarrow x=\frac{-2}{3}$

Here, the value of x is not unique, so option B is correct.

Question 10. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

(A) 27
(B) 18
(C) 81
(D) 512

Answer:

Total number of elements in a 3 × 3 matrix

$=3\times 3=9$

If each entry is 0 or 1 then for every entry there are 2 permutations.

The total permutations for 9 elements

$=2^{9}=512$

Thus, option (D) is correct.

More About NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.1:-

There are 5 solved examples given before this exercise that you can solve. Solving these examples will help you to get conceptual clarity. There are 7 long answer types questions and 3 multiple choice types questions given in the NCERT Solutions for Class 12 Maths Chapter 3 exercise 3.1. These questions are very basic questions based on the matrix and order of the matrix. There are some questions in Class 12th Maths chapter 3 exercise 3.1 based on the equality of the matrices of order 2x2 and 3x3.

Also Read| Matrices Class 12 Maths Chapter Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.1:-

As most of the questions in the board exams are directly asked from the NCERT textbook, NCERT solutions for Class 12 Maths chapter 3 exercise 3.1 becomes very important for the students to solve the NCERT problems.
You are advised to solve the CBSE board's previous year's paper to get familiar with the exam pattern.
Solving more problems will help students to get conceptual clarity.
Class 12 Maths chapter 3 exercise 3.1 solutions are created based on the CBSE guideline which you can rely upon.

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Key Features Of NCERT Solutions for Exercise 3.1 Class 12 Maths Chapter 3

Comprehensive Coverage: The solutions encompass all the topics covered in ex 3.1 class 12, ensuring a thorough understanding of the concepts.
Step-by-Step Solutions: In this class 12 maths ex 3.1, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
Accuracy and Clarity: Solutions for class 12 ex 3.1 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
Conceptual Clarity: In this 12th class maths exercise 3.1 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
Inclusive Approach: Solutions for ex 3.1 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
Relevance to Curriculum: The solutions for class 12 maths ex 3.1 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

Also see-

NCERT solutions of class 12 subject wise

Subject wise NCERT Exampler solutions

Happy learning!!!

Frequently Asked Questions (FAQs)

1. What is the matrix ?

Concept related to matrices are discussed in ex 3.1 class 12. A matrix is a rectangular array of numbers or functions. The concept of matrix is discussed in the Class 12 Mathematics NCERT textbook. Practice class 12 ex 3.1 exercise to command concepts.

2. What is the order of the matrix ?

If a matrix has m rows and n columns then its order is m x n.

3. what is a row matrix ?

If a matrix has only one row it's called a row matrix. This concept is discussed in the NCERT syllabus of Class 12 Maths

4. what is a column matrix ?

If a matrix has only one column it's called a column matrix.

5. what is a square matrix ?

If a matrix has equal numbers of rows and columns then it's called a square matrix.

6. what is a diagonal matrix ?

If all the non-diagonal elements of a matrix are zero it's called a diagonal matrix.

7. What is a scalar matrix ?

A scalar matrix is a diagonal matrix that has equal diagonal elements.

8. What is identity matrix ?

A square matrix that has all the non-diagonal elements are zero and its diagonal elements are 1 is called an identity matrix.

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quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.

Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!

Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.

If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.

Let me know if you need any other tips for your math prep. Good luck with your studies!

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Jefferson 25 September,2024

i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.

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Kanishka kaushikii 17 September,2024

i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

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Yuvan 30 August,2024

i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

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Ashvadha 12 July,2024

Kya mujhe class 12 ka important question milega jo exam mai per year puche jate hai

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

Thank you and wishing you all the best for your bright future.

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kumardurgesh1802 10 July,2024

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Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

Option 1) less than 3	Option 2) more than 3 but less than 6
Option 3) more than 6 but less than 9	Option 4) more than 9

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