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. I m portant topics that are going to be discus sed in this chapter are matrices, the order of a matrix, types of matrices, equality of matrices, operations like addition multiplication on matrices, symmetric and skewsymmetric 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. In this chapter , there are 4 exercises & a miscellaneous exercise.
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 and also fto practice more, at the end of the chapter, 15 questions are given in the miscellaneous exercise.
Topics and subtopics 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 SkewSymmetric 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:
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.
CBSE 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 (BC). 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 righthand side not equal to the lefthand 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 (3000x)
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 (3000x)
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.
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:
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 skewsymmetric matrix.
Answer:
The transpose of A is
Since, so given matrix is a skewsymmetric 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 skewsymmetric 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 skewsymmetric 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 skewsymmetric 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,( ABBA)' 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 .
Solutions of NCERT 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,( ABBA)' 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 skewsymmetric matrix, then .
Replace by  ,
i.e. .
Thus, if A is a skewsymmetric matrix then is a skewsymmetric matrix.
Hence, the matrix B′AB is symmetric or skewsymmetric according to as A is symmetric or skewsymmetric.
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 skewsymmetric, 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 skewsymmetric, 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 chapterwise
chapter 1 
Solutions of NCERT for class 12 maths chapter 1 Relations and Functions 
chapter 2 
CBSE NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions 
chapter 3 
CBSE NCERT Solutions for class 12 maths chapter 3 Matrices 
chapter 4 
Solutions of NCERT for class 12 maths chapter 4 Determinants 
chapter 5 
CBSE NCERT solutions for class 12 maths chapter 5 Continuity and Differentiability 
chapter 6 
NCERT solutions for class 12 maths chapter 6 Application of Derivatives 
chapter 7 

chapter 8 
CBSE NCERT solutions for class 12 maths chapter 8 Application of Integrals 
chapter 9 
NCERT solutions for class 12 maths chapter 9 Differential Equations 
chapter 10 
Solutions of NCERT for class 12 maths chapter 10 Vector Algebra 
chapter 11 
CBSE NCERT solutions for class 12 maths chapter 11 Three Dimensional Geometry 
chapter 12 
NCERT solutions for class 12 maths chapter 12 Linear Programming 
chapter 13 
Solutions of NCERT for class 12 maths chapter 13 Probability 
NCERT solutions for class 12
NCERT Solutions
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 timeconsuming, 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 applicationrelated 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 !!!