# 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 solutions of NCERT 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 CBSE 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.

The topic algebra which contains two topics matrices and determinants which has 13 % weightage in the maths CBSE 12th 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 12th  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 and also fto 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

3.4.2 Multiplication of a matrix by a scalar

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- Solved exercise questions

### Solutions of NCERT class 12 maths chapter 3 Matrices: Exercise 3.1

Question:1(i). In the matrix , write:

The order of the matrix

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

Question 1(ii). In the matrix , write:

The number of elements

(ii) The number of elements .

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

Write the elements a13, a21, a33a24, a23

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

A  matrix has 24 elements.

The possible orders are :

.

If it has 13 elements, then possible orders are :

.

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:

(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:

A  2 × 2 matrix,

(ii)

Hence, the matrix is

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

(iii)

Hence, the matrix is given by

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

(i)

Hence, the required matrix of the given order is

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

A    3 × 4   matrix,

(ii)

Hence, the matrix is

(i)

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

(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

(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

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

(A)

(B)

(C)

(D)    None of these

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.

(A)

(B)    Not possible to find

(C)

(D)

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.

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

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.

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

Question 1(i) Let

Find each of the following:

A + B

(i) A + B

The addition of matrix can be done as follows

Question 1(ii)    Let

Find each of the following:

A - B

(ii) A - B

Question 1(iii) Let

Find each of the following:

3A - C

(iii) 3A - C

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

Question 1(iv) Let

Find each of the following:

AB

(iv) AB

Question 1(v)    Let

Find each of the following:

BA

The multiplication is performed as follows

,

Question 2(i). Compute the following:

(i)

Question 2(ii). Compute the following:

(ii) The addition operation can be performed as follows

Question 2(iii).  Compute the following:

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

Question 2(iv).    Compute the following:

(iv) the addition is done as follows

since

Question 3(i). Compute the indicated products.

(i) The multiplication is performed as follows

Question 3(ii). Compute the indicated products.

(ii) the multiplication can be performed as follows

Question 3(iii). Compute the indicated products.

(iii) The multiplication can be performed as follows

Question 3(iv). Compute the indicated products.

(iv) The multiplication is performed as follows

Question 3(v). Compute the indicated products.

(v)  The product  can be computed as follows

Question 3(vi). Compute the indicated products.

(vi) The given product can be computed as follows

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 5. If  and , then compute 3A - 5B

and

Question 6. Simplify .

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

and

(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

and

(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 8. Find X, if  and

Substituting the value of Y in the above equation

Question 9. Find x and y, if

Now equating LHS and RHS we can write the following equations

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

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.

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.

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

Thus, we have

Put the value of x

Hence, we have

Question 13. If , show that .

To prove :

Hence, we have L.H.S. = R.H.S i.e. .

Question 14(i). Show that

To prove:

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

Question 14(ii). Show that

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

Hence,    i.e. .

Question 15. Find, if

First, we will find ou the value of the square of matrix A

Question 16. If   prove that .

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  17. If  and , find k so that .

We have,

Hence, the value of k is 1.

To prove :

L.H.S :

R.H.S :

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

i.e. .

Rs. 1800

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.

Rs. 2000

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.

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)

(B) k is arbitrary,

(C) p is arbitrary,

(D)

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

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.

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

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

The transpose of the given matrix is

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

interchanging the rows and columns of the matrix A we get

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

Transpose is obtained by interchanging the rows and columns of matrix

Question 2(i).    If  and , then verify

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

and

L.H.S :

R.H.S :

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

.

Question 3(i).    If   and , then verify

To prove:

R.H.S:

Hence, L.H.S = R.H.S i.e. .

Question 3(ii).    If  and , then verify

To prove:

R.H.S:

Hence, L.H.S = R.H.S i.e. .

Question 4.  If  and , then find

:

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

,

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

To prove :

Heence, L.H.S =R.H.S  i.e..

Question 6(i).   If , then verify that

By interchanging rows and columns we get transpose of A

To prove:

L.H.S :

Question 6(ii).  If , then verify that

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.

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.

The transpose of A is

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

Question 8(i). For the matrix , verify that

is a symmetric matrix.

We have

Hence ,   is a symmetric matrix.

Question 8(ii) For the matrix , verify that

is a skew symmetric matrix.

We have

Hence ,   is a skew-symmetric matrix.

Question 9. Find  and , when

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

Let

Thus,    is a symmetric matrix.

Let