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# NCERT Solutions for Class 12 Maths Chapter 3 - Exercise Last updated date: 29th Nov 2023
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## NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.1 (Ex 3.1)

Vedantu provides simple-to-understand and well-crafted NCERT Solutions for Exercise 3.1 Class 12 Maths Matrices. These solutions cover all important topics in the ex 3.1 Class 12 Maths with detailed explanations. Since our experts understand the importance of Maths examination well in Class 12th; our experts strive their best to help students like you to understand the complete Matrices concepts. Our NCERT Class 12 Maths Exercise 3.1 solutions will prove to be most helpful to you in your assignments, preparations for CBSE board examinations and practice sessions.

## 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 $3\times 4$ 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 $A=\left[ \begin{matrix} 1 & 0 & -1 & -2 \\ 3 & 2 & 1 & 0 \\ 5 & 4 & 3 & 2 \\ \end{matrix} \right]$.

5. 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,

$4+z=7$

$\Rightarrow z=3$

$x+z=5$

$\Rightarrow x=2$

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

6. 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),

$3\times 3+d=13$

$\Rightarrow d=4$

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

7. $A={{\left[ {{a}_{y}} \right]}_{m\times n}}$ 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.

8. 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).

9. 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 ${{2}^{9}}=512$.

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NCERT Solutions for Class 12 Maths Chapter 3 Ex 3.1 are available in PDF format. You can access these CBSE Textbook solutions for Exercise 3.1 Class 12 Maths at any time for free. The question wise answers for the Exercise questions can be very helpful for you at the time of CBSE board exams.

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Solutions for Exercise 3.1 Maths Class 12 Matrices are provided in details to make it easy for you to understand the complete concept in a simpler way. Since Matrices is one important Chapter that holds great importance in the 12th board examination, so it becomes vital to have a thorough and comprehensive understanding of the topic. This way, you will be able to score better and higher in your 12th Class CBSE board examinations.

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Matrices is such an important Chapter in the NCERT books of Class 12th. Mainly, matrices are used to represent the linear transformation from and to the vector field. A rectangular array of a x b numbers (be it complex or real) in the way of horizontal lines (known as rows) and b number of vertical lines (known as rows), is known to be the matrix of order a by b, written as a x b matrix. Usually, this type of array is enclosed by ( ) or [ ]. Remember - Matrices is a very important part in the CBSE exams, which play a crucial factor in various aspects!

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### Important Topics Covered under Class 12th NCERT Maths Chapter 3 Matrices Exercise 3.1

Chapter 3 - Matrices is a very important chapter in Class 12 Maths syllabus. NCERT Solutions for Class 12 Maths Chapter 3 Matrices (Exercise 3.1) mainly focus on the below concepts:

• Meaning of a Matrix: A matrix can be defined as an ordered rectangular array of numbers or elements.

• The representation of elements in a matrix.

• The Order of a Matrix: The order of a matrix can be represented as the number of rows and columns of that particular matrix.

• Types of Matrices: Row matrix, column matrix, diagonal matrix, square matrix, identity matrix, scalar matrix, and zero matrix.

• The Equality of Matrices: If two matrices have the same order or dimension and the elements of the two matrices are equal, then the two matrices are called equal matrices.

This exercise consists of questions such as constructing a new matrix for a given element, finding orders of a matrix, etc.

### Importance of Class 12 Maths Chapter 3 Matrix

Learning about the core concept of the matrix is very crucial in the studies of Mathematics, Physics (electrical circuits, optics, and quantum mechanics), and other subjects and also in several aspects of real life. The concept of the matrix is used for coding and encrypting messages in the field of programming. To track user information, manage databases, and perform search queries, many IT companies use matrices as data structures. Matrices are used for representing real-world data such as the population of people, mortality rates, infertility rates, etc.

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### Class 12 Maths Chapter 3 Exercises

 Chapter 3 - Matrix Exercises in PDF Format Exercise 3.1 10 Questions & Solutions (5 Short Answers, 5 Long Answers) 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 18 Questions & Solutions (18 Short Answers)

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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

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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.