# NCERT Solutions for Class 12 Maths Chapter 3 Matrices

**NCERT Solutions for Class 12 Maths Chapter 3 Matrices: **Matrix is a Latin word which means "womb", an environment where something grows. In this article, you will find NCERT solutions for class 12 maths chapter 3 matrices. Matrix is an array of numbers or mathematical objects for which some operations like addition and multiplication are defined. Matrices is an important and powerful tool in mathematics and it's basically introduced to solve simultaneous linear equations. Matrices have a lot of applications like solving the system of linear equations, doing transformations of one vector space to another, etc. It has applications in engineering, for example, the image or video that you see are matrices of color intensities. In this chapter, you will learn about matrices and it's properties. In order to get the NCERT solutions for class 12 maths chapter 3 matrices, you can go through this article. Important topics that are going to be discussed in this chapter are matrices, the order of a matrix, types of matrices, equality of matrices, operations like addition multiplication on matrices, symmetric and skew-symmetric matrices, etc. In NCERT solutions for class 12 maths chapter 3 matrices article, questions from all topics are covered. The practice is very important to command over any chapter of CBSE maths, so you should try to solve every problem on your own. If you are not able to solve, you can take the help of NCERT solutions for class 12 maths chapter 3 matrices, which are explained in a detailed manner. Check all ** NCERT solutions ** from class 6 to 12 at a single place which will be helpful in order to learn CBSE science and maths. Here you will get NCERT solutions for class 12 also.

The topic algebra which contains two topics matrices and determinants which has 13 % weightage in the maths CBSE 12^{th }board final examination, which means you will see 10 marks questions from these two chapters matrices, and determinants in 12th board final exam out of 80 marks. Matrix is a very important chapter from the exam point of view, also from the application point of view, it is very important in further studies like engineering. In this chapter, there are 4 exercises with 62 questions. All these questions are prepared and explained in the NCERT solutions for class 12 maths chapter 3 matrices article. These solutions of NCERT will help you to understand the concept more easily, and perform well in the CBSE 12^{th } board exam.

**What ****are ****matrices? **

Matrix is an array of numbers. Matrix is a mode of representing data to ease calculation and it is one of the most important tools of mathematics because matrices simplify our work to a great extent when compared with other straight forward methods. Matrices are used as a representation of the coefficients in the system of linear equations, electronic spreadsheet programs, also used in business and science. For the students to understand NCERT class 12 maths chapter 3 matrices in a better way total of 28 solved examples are given to practice more, at the end of the chapter, 15 questions are given in the miscellaneous exercise.

** Topics and sub-topics of NCERT Grade 12 Maths Chapter 3 Matrices **

3.1 Introduction

3.2 Matrix

3.2.1 Order of a matrix

3.3 Types of Matrices

3.3.1 Equality of matrices

3.4 Operations on Matrices

3.4.1 Addition of matrices

3.4.2 Multiplication of a matrix by a scalar

3.4.3 Properties of matrix addition

3.4.4 Properties of scalar multiplication of a matrix

3.4.5 Multiplication of matrices

3.4.6 Properties of multiplication of matrices

3.5. Transpose of a Matrix

3.5.1 Properties of the transpose of the matrices

3.6 Symmetric and Skew-Symmetric Matrices

3.7 Elementary Operation (Transformation) of a Matrix

3.8 Invertible Matrices

3.8.1 Inverse of a matrix by elementary operations

## ** NCERT solutions for class 12 maths chapter 3 Matrices: Exercise 3.1 **

** Question:1(i). ** In the matrix , write:

** Answer: **

(i) The order of the matrix = number of row number of columns .

** Question 1(iii). ** In the matrix , write:

Write the elements * a _{ 13 } * ,

*a*,

_{ 21 }*a*

_{ 33 },*a*,

_{ 24 }*a*

_{ 23 } ** Answer: **

(iii) An element implies the element in raw number i and column number j.

* _{ } *

** 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 :

.

If it has 13 elements, then possible orders are :

.

** 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

If it has 5 elements, then possible orders are :

.

** Question 4(i). ** Construct a 2 × 2 matrix, whose elements are given by:

** Answer: **

(i)

Each element of this matrix is calculated as follows

Matrix A is given by

** Question 4(ii). ** Construct a 2 × 2 matrix, , whose elements are given by:

** Answer: **

A 2 × 2 matrix,

(ii)

Hence, the matrix is

** Question 4(iii). ** Construct a 2 × 2 matrix, , whose elements are given by:

** Answer: **

(iii)

Hence, the matrix is given by

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

** Answer: **

(i)

Hence, the required matrix of the given order is

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

** Answer: **

A 3 × 4 matrix,

(ii)

Hence, the matrix is

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

** Answer: **

(i)

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

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

** Answer: **

(ii)

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

Solving equation (i) and (ii) ,

solving this equation we get,

Putting the values of y, we get

And also equating the first element of the second raw

,

Hence,

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

** Answer: **

(iii)

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

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,

, and

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

** Answer: **

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

Solving equation 1 and 3 , we get

Putting the value of a in equation 2, we get

Putting the value of c in equation 4 , we get

** Question 8. ** is a square matrix, if

** Answer: **

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

Thus, for ** ** to be a square matrix m and n should be equal.

Option (c) is correct.

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

** Answer: **

Given,

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

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:

** Answer: **

Total number of elements in a 3 × 3 matrix

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

The total permutations for 9 elements

Thus, option (D) is correct.

## **NCERT solutions for class 12 maths chapter 3 Matrices: Exercise 3.2 **

** Question 1(i) ** Let , ,

** Answer: **

(i) A + B

The addition of matrix can be done as follows

** Question 1(iii) ** Let , ,

** Answer: **

(iii) 3A - C

First multiply each element of A with 3 and then subtract C

** Question 2(ii). ** Compute the following:

** Answer: **

(ii) The addition operation can be performed as follows

** Question 2(iii). ** Compute the following:

** Answer: **

(iii) The addition of given three by three matrix is performed as follows

** Question 3(i). ** Compute the indicated products.

** Answer: **

(i) The multiplication is performed as follows

** Question 3(ii). ** Compute the indicated products.

** Answer: **

(ii) the multiplication can be performed as follows

** Question 3(iii). ** Compute the indicated products.

** Answer: **

(iii) The multiplication can be performed as follows

** Question 3(iv). ** Compute the indicated products.

** Answer: **

(iv) The multiplication is performed as follows

** Question 3(vi). ** Compute the indicated products.

** Answer: **

(vi) The given product can be computed as follows

** Question 4. ** If , and , then compute (A+B) and (B-C). Also verify that A + (B - C) = (A + B) - C

** Answer: **

, and

Now, to prove A + (B - C) = (A + B) - C

(Puting value of from above)

Hence, we can see L.H.S = R.H.S =

** Question 6. ** Simplify .

** Answer: **

The simplification is explained in the following step

the final answer is an identity matrix of order 2

** Question 7(i). ** Find X and Y, if

** Answer: **

(i) The given matrices are

and

Adding equation 1 and 2, we get

Putting the value of X in equation 1, we get

** Question 7(ii). ** Find X and Y, if

** Answer: **

(ii) and

Multiply equation 1 by 3 and equation 2 by 2 and subtract them,

Putting value of Y in equation 1 , we get

** Question 10. ** Solve the equation for x, y, z and t, if

** Answer: **

Multiplying with constant terms and rearranging we can rewrite the matrix as

Dividing by 2 on both sides

** Question 11. ** If , find the values of x and y.

** Answer: **

Adding both the matrix in LHS and rewriting

Adding equation 1 and 2, we get

Put the value of x in equation 2, we have

** Question 12. ** Given , find the values of x, y, z and w.

** Answer: **

If two matrices are equal than corresponding elements are also equal.

Thus, we have

Put the value of x

Hence, we have

** Question 14(i). ** Show that

** Answer: **

To prove:

Hence, the right-hand side not equal to the left-hand side, that is

** Question 14(ii). ** Show that

** Answer: **

To prove the following multiplication of three by three matrices are not equal

Hence, i.e. .

** Question 16. ** If prove that .

** Answer: **

First, find the square of matrix A and then multiply it with A to get the cube of matrix A

L.H.S :

Hence, L.H.S = R.H.S

i.e. .

** Question 18. ** If and I is the identity matrix of order 2, show that

** Answer: **

To prove :

L.H.S :

R.H.S :

Hence, we can see L.H.S = R.H.S

i.e. .

** Answer: **

Let Rs. x be invested in the first bond.

Money invested in second bond = Rs (3000-x)

The first bond pays 5% interest per year and the second bond pays 7% interest per year.

To obtain an annual total interest of Rs. 1800, we have

(simple interest for 1 year )

Thus, to obtain an annual total interest of Rs. 1800, the trust fund should invest Rs 15000 in the first bond and Rs 15000 in the second bond.

** Answer: **

Let Rs. x be invested in the first bond.

Money invested in second bond = Rs (3000-x)

The first bond pays 5% interest per year and the second bond pays 7% interest per year.

To obtain an annual total interest of Rs. 1800, we have

(simple interest for 1 year )

Thus, to obtain an annual total interest of Rs. 2000, the trust fund should invest Rs 5000 in the first bond and Rs 25000 in the second bond.

** Answer: **

The bookshop has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books.

Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively.

The total amount the bookshop will receive from selling all the books:

The total amount the bookshop will receive from selling all the books is 20160.

** Q21. ** The restriction on n, k and p so that PY + WY will be defined are:

(A)

** Answer: **

P and Y are of order and respectivly.

PY will be defined only if k=3, i.e. order of PY is .

W and Y are of order and respectivly.

WY is defined because the number of columns of W is equal to the number of rows of Y which is 3, i.e. the order of WY is

Matrices PY and WY can only be added if they both have same order i.e = implies p=n.

Thus, are restrictions on n, k and p so that PY + WY will be defined.

Option (A) is correct.

** Question 22 ** Assume X, Y, Z, W and P are matrices of order * 2 × n, 3 × k, 2 × p, n × 3 * and * p × k * ,

respectively. Choose the correct answer in Exercises 21 and 22. ** ** If * n = p * , then the order of the matrix is:

(A) * p × 2 *

(B) * 2 × n *

(C) * n × 3 *

(D) * p × n *

** Answer: **

X has of order .

7X also has of order .

Z has of order .

5Z also has of order .

Mtarices 7X and 5Z can only be subtracted if they both have same order i.e = and it is given that p=n.

We can say that both matrices have order of .

Thus, order of is .

Option (B) is correct.

**NCERT Solutions for class 12 maths chapter -3 Matrices: Exercise 3.3 **

** Question 1(i). ** Find the transpose of each of the following matrices:

** Answer: **

The transpose of the given matrix is

** Question 1(ii). ** Find the transpose of each of the following matrices:

** Answer: **

interchanging the rows and columns of the matrix A we get

** Question 1(iii) ** Find the transpose of each of the following matrices:

** Answer: **

Transpose is obtained by interchanging the rows and columns of matrix

** Question 2(i). ** If and , then verify

** Answer: **

and

L.H.S :

R.H.S :

Thus we find that the LHS is equal to RHS and hence verified.

** Question 2(ii). ** If and , then verify

** Answer: **

and

L.H.S :

R.H.S :

Hence, L.H.S = R.H.S. so verified that

.

** Question 4. ** If and , then find

** Answer: **

:

Transpose is obtained by interchanging rows and columns and the transpose of A+2B is

** Question 5(i) ** For the matrices A and B, verify that , where

** Answer: **

,

To prove :

Hence, L.H.S =R.H.S

so it is verified that .

** Question 5(ii) ** For the matrices A and B, verify that , where

** Answer: **

,

To prove :

Heence, L.H.S =R.H.S i.e. .

** Question 6(i). ** If , then verify that

** Answer: **

By interchanging rows and columns we get transpose of A

To prove:

L.H.S :

** Question 6(ii). ** If , then verify that

** Answer: **

By interchanging columns and rows of the matrix A we get the transpose of A

To prove:

L.H.S :

** Question 7(i). ** Show that the matrix is a symmetric matrix.

** Answer: **

the transpose of A is

Since, so given matrix is a symmetric matrix.

** Question 7(ii) ** Show that the matrix is a skew-symmetric matrix.

** Answer: **

The transpose of A is

Since, so given matrix is a skew-symmetric matrix.

** Question 9. ** Find and , when

** Answer: **

the transpose of the matrix is obtained by interchanging rows and columns

** Question 10(i). ** Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

** Answer: **

Let

Thus, is a symmetric matrix.

Let

Thus, is a skew symmetric matrix.

Represent A as sum of B and C.

** Question:10(ii). ** Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

** Answer: **

Let

Thus, is a symmetric matrix.

Let

Thus, is a skew-symmetric matrix.

Represent A as the sum of B and C.

** Question 10(iii). ** Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

** Answer: **

Let

Thus, is a symmetric matrix.

Let

Thus, is a skew-symmetric matrix.

Represent A as the sum of B and C.

** Question 10(iv). ** Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

** Answer: **

Let

Thus, is a symmetric matrix.

Let

Thus, is a skew-symmetric matrix.

Represent A as the sum of B and C.

** Question 11 ** Choose the correct answer in the Exercises 11 and 12.

** ** If A, B are symmetric matrices of same order, then AB – BA is a

(A) Skew symmetric matrix

(B) Symmetric matrix

(C) Zero matrix

(D) Identity matrix

** Answer: **

** ** If A, B are symmetric matrices then

and

we have,

Hence, we have

Thus,( AB-BA)' is skew symmetric.

Option A is correct.

## ** NCERT solutions for class 12 maths chapter 3 Matrices: Exercise 3.4 **

** Answer: **

Use the elementary transformation we can find the inverse as follows

** Answer: **

Thus we have obtained the inverse of the given matrix through elementary transformation

** Answer: **

Using elementary transformations

.

** Answer: **

** Answer: **

.

Thus the inverse of matrix A is obtained.

** Answer: **

Use the elementary transformation

** **

** **

** **

** **

.

** Answer: **

.

Thus the inverse of matrix A is obtained using elementary transformation.

** Answer: **

Thus using elementary transformation inverse of A is obtained as

.

** Answer: **

Thus using elementary transformation the inverse of A is obtained as

.

** Answer: **

.

Thus the inverse of A is obtained using elementary transformation.

** Answer: **

thus the inverse of matrix A is

.

** Answer: **

Hence, we can see all the zeros in the second row of the matrix in L.H.S so does not exist.

** Answer: **

** **

** **

** **

** **

so the inverse of matrix A is

.

** Answer: **

Hence, we can see all upper values of matirix are zeros in L.H.S so does not exists.

** Answer: **

Thos the Inverse of A is

. .

** Answer: **

and

and

and

Thus the inverse of three by three matrix A is

. .

** Answer: **

and

Thus the inverse of A is obtained as

. .

** Question:18 ** Matrices A and B will be inverse of each other only if

** Answer: **

We know that if A is a square matrix of order n and there is another matrix B of same order n, such that , then B is inverse of matrix A.

In this case, it is clear that A is inverse of B.

Hence, ** m ** atrices A and B will be inverse of each other only if .

Option D is correct .

## ** NCERT solutions for class 12 maths chapter -3 Matrices: Miscellaneous exercise **

** Question:1 ** Let , show that , where I is the identity matrix of order 2 and .

** Answer: **

Given :

To prove :

For n=1,

The result is true for n=1.

Let result be true for n=k,

Now, we prove that the result is true for n=k+1,

Put the value of in above equation,

Hence, the result is true for n=k+1.

Thus, we have where , .

** Question 2. ** If then show that , .

** Answer: **

Given :

To prove:

For n=1, we have

Thus, the result is true for n=1.

Now, take n=k,

For, n=k+1,

Thus, the result is true for n=k+1.

Hence, we have , where .

** Question 3. ** If , then prove that , where * n * is any positive integer.

** Answer: **

Given :

To prove:

For n=1, we have

Thus, result is true for n=1.

Now, take result is true for n=k,

For, n=k+1,

Thus, the result is true for n=k+1.

Hence, we have , where .

** Question 4. ** If A and B are symmetric matrices, prove that is a skew symmetric matrix.

** Answer: **

If A, B are symmetric matrices then

and

we have,

Hence, we have

Thus,( AB-BA)' is skew symmetric.

** Question 5. ** Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

** Answer: **

Let be a A is symmetric matrix , then

Consider,

Replace by

i.e.

Thus, if A is a symmetric matrix than is a symmetric matrix.

Now, let A be a skew-symmetric matrix, then .

Replace by - ,

i.e. .

Thus, if A is a skew-symmetric matrix then is a skew-symmetric matrix.

Hence, the matrix B′AB is symmetric or skew-symmetric according to as A is symmetric or skew-symmetric.

** Question 6. ** Find the values of * x * , * y * , * z * if the matrix satisfy the equation

** Answer: **

Thus equating the terms elementwise

** Question 10(a) ** A manufacturer produces three products x, y, z which he sells in two markets.

Annual sales are indicated below:

** Market Products **

I 10,000 2,000 18,000

II 6,000 20,000 8,000

** Answer: **

The unit sale prices of x, y and z are ` 2.50, ` 1.50 and ` 1.00, respectively.

The total revenue in the market I with the help of matrix algebra can be represented as :

The total revenue in market II with the help of matrix algebra can be represented as :

Hence, total revenue in the market I is 46000 and total revenue in market II is 53000.

** Question 10(b). ** A manufacturer produces three products x, y, z which he sells in two markets.

Annual sales are indicated below:

** Market Products **

I 10,000 2,000 18,000

II 6,000 20,000 8,000

** Answer: **

The unit costs of the above three commodities are ` 2.00, ` 1.00 and 50 paise respectively.

The total cost price in market I with the help of matrix algebra can be represented as :

Total revenue in the market I is 46000 , gross profit in the market is

The total cost price in market II with the help of matrix algebra can be represented as :

Total revenue in market II is 53000, gross profit in the market is

** Question 11. ** Find the matrix * X * so that

** Answer: **

The matrix given on R.H.S is matrix and on LH.S is matrix.Therefore, X has to be matrix.

Let X be

Taking,

Hence, we have

Matrix X is .

** Answer: **

A and B are square matrices of the same order such that ,

To prove : ,

For n=1, we have

Thus, the result is true for n=1.

Let the result be true for n=k,then we have

Now, taking n=k+1 , we have

Thus, the result is true for n=k+1.

Hence, we have , .

To prove:

For n=1, we have

Thus, the result is true for n=1.

Let the result be true for n=k,then we have

Now, taking n=k+1 , we have

Thus, the result is true for n=k+1.

Hence, we have and for all .

** Question 13 ** Choose the correct answer in the following questions:

** Answer: **

Thus we obtained that

Option C is correct.

** Question 14. ** If the matrix A is both symmetric and skew-symmetric, then

(A) A is a diagonal matrix

(B) A is a zero matrix

(C) A is a square matrix

(D) None of these

** Answer: **

** ** If the matrix A is both symmetric and skew-symmetric, then

and

Hence, A is a zero matrix.

Option B is correct.

** Question 15. ** If A is square matrix such that , then is equal to

** Answer: **

A is a square matrix such that

(Replace by )

Hence, we have

Option C is correct.

** NCERT solutions for class 12 maths - Chapter wise **

** NCERT solutions for class 12 subject wise **

** NCERT Solutions class wise **

**NCERT solutions for class 12****NCERT solutions for class 11****NCERT solutions for class 10****NCERT solutions for class 9**

** Some applications of matrices: **

- If you have to solve two equations with two variables, you can solve by elimination or substitution method but if you have 10 equations with 10 variables to solve, then the above method will be tough and time-consuming, you can use the matrix to solve the system of linear equations. In the NCERT solutions for class 12 maths chapter 3 matrices, you will find some application-related problems also.
- In the storage of huge data in the computer, matrices make it easy
- Any image or video that we see is also matrices of color intensities.
- Its applications are in every branch of physics, scientific studies, probability theory, statistics, stochastic process, etc.
- Solutions of NCERT for class 12 maths chapter 3 matrices will build your basic understanding of matrix which is required to study any branch of engineering

** Happy learning !!! **

## Frequently Asked Question (FAQs) - NCERT Solutions for Class 12 Maths Chapter 3 Matrices

**Question: **How the NCERT solutions are helpful in the board exam ?

**Answer: **

NCERT solutions are not only important when you stuck while solving the problems but you will get to know how to answer in the board exam in order to get good marks in the board exam.

**Question: **What are the important topics in chapter matrices ?

**Answer: **

Matrices, order of a matrix, types of matrices, equality of matrices, operations like addition multiplication on matrices, symmetric and skew-symmetric matrices are the important topics in this chapter.

**Question: **Does CBSE provides the solutions of NCERT class 12 maths ?

**Answer: **

No, CBSE doesn’t provided NCERT solutions for any class or subject.

**Question: **Where can I find the complete solutions of NCERT class 12 maths ?

**Answer: **

A Here you will get the detailed NCERT solutions for class 12 maths by clicking on the link.

**Question: **What is the weightage of the chapter matrices for CBSE board exam ?

**Answer: **

The topic algebra which contains two topics matrices and determinants which has 13 % weightage in the maths CBSE 12^{th }board final examination.

**Question: **Which is the official website of NCERT ?

**Answer: **

http://ncert.nic.in/ is the official website of the NCERT where you can get NCERT textbooks and syllabus from class 1 to 12 for all the subjects.

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