NCERT Solutions For Class 12 Maths Chapter 3 – Matrices Class 12 NCERT Solutions

NCERT Solutions For Class 12 Maths Chapter 3 covers all the exercises in the simplest method, which are solved by India’s best teachers. Students who are searching for Matrices Class 12 NCERT Solutions can easily approach this article. This can help you to solve the questions in an easy manner. Multiple times of practice of Matrices Class 12 notes give you the best result in the exam. The given table contains a list of subchapters of Class 12 Maths Matrices. Each solution was given in a detailed explanation. This can be helpful to get a better score in the examination. Class 12 NCERT Solutions Maths helps to complete your home assignments as well as revision purposes.

NCERT Solutions Of Class 12 Maths Chapter 3 Matrices

Section Name Topic Name
3.1 Introduction
3.1 Matrix
3.3 Types Of Matrices
3.4 Operations on Matrices

 

NCERT Solutions For Class 12 Maths Chapter 3 Matrices Ex 3.1- Introduction

Question 1:
In the matrix NCERT Solutions Class 12 Maths Chapter 3 Martrices Q 1

write:

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(i) The order of the matrix (ii) The number of elements, (iii) Write the elements a13a21a33a24a23
Answer:

(i) In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, the order of the matrix is 3 × 4.

(ii) Since the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.
(iii) a13 = 19, a21 = 35, a33 = −5, a24 = 12, a23 = NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 1(a)

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

We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 24 elements, we have to find all the ordered pairs of natural numbers whose product is 24.

The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and (6, 4)

Hence, the possible orders of a matrix having 24 elements are:

1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, and 6 × 4

(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.

Hence, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.


Question 3:

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Answer:

We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 18 elements, we have to find all the ordered pairs of natural numbers whose product is 18.

The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6,), and (6, 3)

Hence, the possible orders of a matrix having 18 elements are:

1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, and 6 × 3

(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.

Hence, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.


Question 4:
Construct a 2 × 2 matrix, NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4, whose elements are given by:
(i) NCERT Solutions Class 12 Maths Chapter3 Ex 3.1 Q 4(a)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(b)
(iii) NCERT Solutions class 12 Maths Chapter 3 Ex 3.1 Q 4(j)

Answer:
In general, a 2 × 2 matrix is given by NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(c)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(e)
Therefore, the required matrix is NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(f)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(g)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(h)
Therefore, the required matrix is NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(i)
(iii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(k)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 4(l)
Therefore, the required matrix is NCERT Solutions Class 12 Maths chapter 3 Ex 3.1 Q 4(m)


Question 5:

Construct a 3 × 4 matrix, whose elements are given by
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(a)
Answer:
In general, a 3 × 4 matrix is given by NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(b)
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(c)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(d)
Therefore, the required matrix is NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(e)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(f)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(g)
Therefore, the required matrix is NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 5(h)


Question 6:

Find the value of xy, and z from the following equation:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 6 (ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 6(a)
(iii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 6(b)
Answer:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 6

As the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

x = 1, y = 4, and z = 3

(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 6(a)

As the given matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

x + y = 6, xy = 8, 5 + = 5

Now, 5 + z = 5 ⇒ z = 0

We know that:

(x − y)2 = (x + y)2 − 4xy

⇒ (x − y)2 = 36 − 32 = 4

⇒ x − y = ±2

Now, when x − y = 2 and x + y = 6, we get x = 4 and y = 2

When x − = − 2 and x + y = 6, we get x = 2 and = 4

x = 4, y = 2, and z = 0 or x = 2, y = 4, and z = 0

(iii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 6(b)

As the two matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

x + y + z = 9 … (1)

x + z = 5 … (2)

y + z = 7 … (3)

From (1) and (2), we have:

+ 5 = 9

⇒ y = 4

Then, from (3), we have:

4 + z = 7

⇒ z = 3

∴ x + z = 5

⇒ x = 2

∴ x = 2, y = 4, and z = 3


Question 7:

Find the value of abc, and d from the equation:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 7
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 7

As the two matrices are equal, their corresponding elements are also equal.

Comparing the corresponding elements, we get:

a − b = −1 … (1)

2a − b = 0 … (2)

2a + c = 5 … (3)

3c + d = 13 … (4)

From (2), we have:

b = 2a

Then, from (1), we have:

− 2a = −1

⇒ a = 1

⇒ b = 2

Now, from (3), we have:

2 ×1 + c = 5

⇒ c = 3

From (4) we have:

3 ×3 + d = 13

⇒ 9 + = 13 ⇒ d = 4

a = 1, b = 2, c = 3, and d = 4


Question 8:
NCERT Solution Class 12 Maths Chapter 3 Ex 3.1 Q 8
is a square matrix, if

(A) m < n

(B) m > n

(C) m = n

(D) None of these
Answer:

The correct answer is C.

It is known that a given matrix is said to be a square matrix if the number of rows is equal to the number of columns.
Therefore, NCERT Solution Class 12 Maths Chapter 3 Ex 3.1 Q 8is a square matrix, if m = n.


Question 9:

Which of the given values of x and y make the following pair of matrices equal
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 9
(A) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 9(a)
(B) Not possible to find

(C) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 9(b)

(D) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 9(c)
Answer:

The correct answer is B.
It is given that NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 9

Equating the corresponding elements, we get:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.1 Q 9(d)

We find that on comparing the corresponding elements of the two matrices, we get two different values of x, which is not possible.

Hence, it is not possible to find the values of x and y for which the given matrices are equal.


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:

The correct answer is D.

The given matrix of the order 3 × 3 has 9 elements and each of these elements can be either 0 or 1.

Now, each of the 9 elements can be filled in two possible ways.

Therefore, by the multiplication principle, the required number of possible matrices is 29 = 512


Class 12 Maths Chapter 3 Matrices NCERT Solutions Ex 3.2 Matrix

Question 1:
Let NCERT Solutions Class 12 Maths chapter 3 Ex 3.2 Q 1
Find each of the following
(i) A+B (ii) A-B (iii) 3A-C (iv) AB (v) BA
Answer:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 1(a)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 1(b)
(iii)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 1(c)
(iv) Matrix A has 2 columns. This number is equal to the number of rows in matrix B. Therefore, AB is defined as:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 1(d)
(v) Matrix B has 2 columns. This number is equal to the number of rows in matrix A. Therefore, BA is defined as:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 1(e)


Question 2:

Compute the following:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(a)(iii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(b)(v) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(c)
Answer:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(d)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(e)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(f)
(iii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(g)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(h)
(v) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 2(i)
NCERT Solutions Class 12 Maths chapter 3 Ex 3.2 Q 2(j)


Question 3:

Compute the indicated products
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(a)
(iii) NCERT Solutions class 12 Maths chapter 3 Ex 3.2 Q 3(b)
(iv) NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 3(c)
(v) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(d)
(vi) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(e)
Answer:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(f)
(ii) NCERT Solutions Class 12 Maths Chapter 3 EX 3.2 Q 3(g)
(iii) NCERT Solutions class 12 Maths chapter 3 Ex 3.2 Q 3(b)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(h)
(iv) NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 3(c)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(i)
(v) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(d)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(j)
(vi) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 3(e)
NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 3(k)


Question 4:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 4
then compute (A+B) and (B-C). Also verify that A+(B-C)=(A+B)-C.
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 4(a)
Hence we have verified that A+(B-C)=(A+B)-C


Question 5:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 5and NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 5(a)then compute 3A-5B.
Answer:
NCERT Solutions Class 12 Maths Chapter 3 eX 3.2 Q 5(b)



Question 6:
Simplify NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 6
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 6(a)



Question 7:
Find X and Y, if
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 7and NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 7(b)
(ii) NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 7(e) and NCERT Solutions Class 12 Maths Chapter 3 ex 3.2 Q 7(d)
Answer:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 7…(1)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 7(b)…(2)

Adding equations (1) and (2), we get:
NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 7(f)
(ii)
NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 7(e)….(3)
NCERT Solutions Class 12 Maths Chapter 3 ex 3.2 Q 7(d)…(4)

Multiplying equation (3) with (2), we get:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 7(g)

Multiplying equation (4) with (3), we get:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 7(h)

From (5) and (6), we have:
NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 7(i)
NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 7(j)
Now,
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 7(k)


Question 8:
Find X, if NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 8(a)and NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 8(b)
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 8(c)



Question 9:
Find x and y, if NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 9
Answer:
NCERT Solutions Class 12 Maths cahpter 3 Ex 3.2 Q 9(a)

Comparing the corresponding elements of these two matrices, we have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 9(b)

x = 3 and y = 3


Question 10:

Solve the equation for xyz, and t if
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 10(a)
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 10(b)

Comparing the corresponding elements of these two matrices, we get:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 10(c)


Question 11:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 11, find values of x and y.
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 11(a)

Comparing the corresponding elements of these two matrices, we get:

2x − y = 10 and 3x + y = 5

Adding these two equations, we have:

5x = 15

⇒ x = 3

Now, 3x + y = 5

⇒ y = 5 − 3x

⇒ y = 5 − 9 = −4

x = 3 and y = −4


Question 12:
Given NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 12find the values of xyz and w.
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 12(a)

Comparing the corresponding elements of these two matrices, we get:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 12(b)


Question 13:
If NCERT Solutions Class 12 Maths chapter 3 Ex 3.2 Q 13Show that NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 13(a)
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 13(b)



Question 14:

Show that
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 14
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 14(a)
Answer:
(i)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 14(b)
(ii)
NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 14(c)


Question 15:
Find NCERT Solutions class 12 Maths Chapter 3 ex 3.2 Q 15if NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 15(b)
Answer:

We have A2 = A × A
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 15(a)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 15(c)



Question 16:
If NCERT Solutions Class 12 maths Chapter 3 ex 3.2 Q 16prove that NCERT Solutions class 12 Maths Chapter 3 Ex 3.2 Q 16(a)
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 16(b)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 16(c)


Question 17:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 17and NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 17(a)find k so that NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 17(b)
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 17(c)

Comparing the corresponding elements, we have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 17(d)

Thus, the value of k is 1.


Question 18:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 18and I is the identity matrix of order 2, show that NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 18(a)
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 18(b)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 18(c)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 18(d)



Question 19:

A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:

(a) Rs 1,800 (b) Rs 2,000
Answer:

(a) Let Rs x be invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 − x).

It is given that the first bond pays 5% interest per year and the second bond pays 7% interest per year.

Therefore, in order to obtain an annual total interest of Rs 1800, we have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 19

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

(b) Let Rs x be invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 − x).

Therefore, in order to obtain an annual total interest of Rs 2000, we have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 19(a)
Thus, in order to obtain an annual total interest of Rs 2000, the trust fund should invest Rs 5000 in the first bond and the remaining Rs 25000 in the second bond.


Question 20:
The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, and 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
Answer:

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

The selling prices of a chemistry book, a physics book, and an economics book are respectively given as Rs 80, Rs 60, and Rs 40.

The total amount of money that will be received from the sale of all these books can be represented in the form of a matrix as:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 20

Thus, the bookshop will receive Rs 20160 from the sale of all these books.


Question 21:
Assume XYZW, and P are matrices of order NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 21,and NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 21(a) respectively. The restriction on nk, and p that NCERT Solutions Class 12 Maths Chapter 3 Ex 3.2 Q 21(b)will be defined are:

A. k = 3, p = n

B. k is arbitrary, p = 2

C. p is arbitrary, k = 3

D. k = 2, p = 3
Answer:

Matrices P and Y are of the orders p × k and 3 × k respectively.

Therefore, matrix PY will be defined if k = 3. Consequently, PY will be of the order p × k.

Matrices W and Y are of the orders × 3 and 3 × k respectively.

Since the number of columns in W is equal to the number of rows in Y, matrix WY is well-defined and is of the order × k.

Matrices PY and WY can be added only when their orders are the same.

However, PY is of the order p × k and WY is of the order n × k. Therefore, we must have p = n.

Thus, = 3 and p = n are the restrictions on nk, and p so that PY+WY will be defined.


Question 22:
Assume XYZW, and P are matrices of order 2×n, 3×k, 2×p, n×3, and p× k respectively. If n = p, then the order of the matrix 7x-5z is A p × 2 B 2 × n C n× 3 D p× n
Answer:

The correct answer is B.

Matrix X is of the order 2 × n.

Therefore, matrix 7X is also of the same order.

Matrix Z is of the order 2 × p, i.e., 2 × n [Since n = p]

Therefore, matrix 5Z is also of the same order.

Now, both the matrices 7X and 5Z are of the order 2 × n.

Thus, matrix 7X − 5Z is well-defined and is of the order 2 × n.


NCERT Solutions For Class 12 Maths Chapter 3 Matrices Ex 3.3 Types of Matrices

Question 1:

Find the transpose of each of the following matrices:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 1(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 1(a)(iii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 1(b)
Answer:
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 1(c)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 1(d)

(iii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 1(e)


Question 2:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 2and NCERT Solutions Class 12 Maths Chapter 3 ex 3.3 Q 2(a)then verify that
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 2(b)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 2(c)
Answer:
We have
NCERT Solutions Class 12 Maths Chapter 3 ex 3.3 Q 2(d)
(i)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 2(e)
(ii)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 2(f)


Question 3:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 3and NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 3(a)then verify that
(i) NCERT Solutions class 12 Maths Chapter 3 Ex 3.3 Q 3(b)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 3(c)
Answer:
(i) It is known that NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 3(d)

Therefore, we have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 3(e)

(ii)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 3(f)


Question 4:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 4and NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 4(a)then find NCERT Solutions class 12 Maths Chapter 3 ex 3.3 Q 4(b)
Answer:
We know that NCERT Solutions Class 12 Maths Chapter 3 ex 3.3 Q 4(c)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 4(d)


Question 5:
For the matrices A and B, verify that (AB)′ = NCERT Solutions class 12 Maths Chapter 3 Ex 3.3 Q 5where
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 5(a)
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 5(b)
Answer:
(i)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 5(c)
(ii)
NCERT Solutions Class 12 Maths Chapter 3 ex 3.3 Q 5(d)


Question 6:
If (i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 6then verify that NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 6(a)
(ii)  NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 6(b)then verify that NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 6(a)
Answer:
(i)
NCERT Solutions class 12 Maths Chapter 3 ex 3.3 Q 6(c)

(ii)
NCERT Solutions Class 12 Maths Chapter 3 ex 3.3 Q 6(d)


Question 7:
(i) Show that the matrix NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 7is a symmetric matrix
(ii) Show that the matrix NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 7(a)is a skew-symmetric matrix
Answer:

(i) We have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 7=A

NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 7(b)
Hence, A is a symmetric matrix.

(ii) We have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 7(c)

Hence, A is a skew-symmetric matrix.


Question 8:
For the matrix NCERT Solutins Class 12 Maths Chapter 3 Ex 3.3 Q 8verify that
(i) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 8(a)is a symmetric matrix
(ii) NCERT Solutions class 12 Maths Chapter 3 Ex 3.3 Q 8(b)is a skew-symmetric matrix
Answer:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 8(c)
(i) NCERT Solutions class 12 Maths chapter 3 Ex 3.3 Q 8(d)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 8(e)
Hence,
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 8(a)is a symmetric matrix.
(ii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 8(f)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 8(g)
Hence,
NCERT Solutions class 12 Maths Chapter 3 Ex 3.3 Q 8(b)is a skew-symmetric matrix.


Question 9:
Find
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 9and NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 9(a)when NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 9(c)
Answer:
The given matrix is NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 9(c)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 9(b)


Question 10:

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
(i) NCERT Solutions Claas 12 Maths Chapter 3 Ex 3.3 Q 10
(ii) NCERT Solutions Class 12 Maths Chapter 3 ex 3.3 Q 10(a)
(iii) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(b)
(iv) NCERT Solutions Class 12 Maths chapter 3 Ex 3.3 Q 10(c)
Answer:
(i)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(d)
Thus NCERT Solutions Class 12 Maths chapter 3 Ex 3.3 Q 10(e) is a symmetric matrix.
NCERT Solutions Class 12 Maths chapter 3 ex 3.3 Q 10(g)
thus NCERT Solutions Class 12 Maths chapter 3 ex 3.3 Q 10(h) is a skew-symmetric matrix.

Representing A as the sum of P and Q:
NCERT Solutions class 12 Maths Chapter 3 Ex 3.3 Q 10(i)
(ii)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(j)
thus NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(k)is a symmetric matrix.
NCERT Solutions Class 12 Maths chapter 3 Ex 3.3 Q 10(l)
thus NCERT Solutions class 12 maths chapter 3 Ex 3.3 Q 10(m)

is a skew-symmetric matrix.

Representing A as the sum of P and Q:
NCERT Solutions class 12 maths Chapter 3 Ex 3.3 Q 10(n)
(iii)
NCERT Solutions Class 12 Maths chapter 3 EX 3.3 Q 10(o)
thus NCERT Solutions Class 12 Maths chapter 3 Ex 3.3 Q 10(p)is a symmetric matrix
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(q)
thus NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(r)

is a skew-symmetric matrix.

Representing A as the sum of P and Q:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(s)
(iv)
NCERT Solutions Class 12 Maths cahpter 3 Ex 3.3 Q 10(t)
thus NCERT Solutions Class 12 Maths chapter 3 Ex 3.3 Q 10(p)is a symmetric matrix.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(t)
thus NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(r)

is a skew-symmetric matrix.

Representing A as the sum of P and Q:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 10(u)


Question 11:

If AB are symmetric matrices of the same order, then AB − BA is a

A. Skew symmetric matrix B. Symmetric matrix

C. Zero matrix D. Identity matrix
Answer:

The correct answer is A.

A and B are symmetric matrices, therefore, we have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 11
Thus, (AB − BA) is a skew-symmetric matrix


Question 12:
If NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 12then NCERT Solutons Class 12 Maths Chapter 3 Ex 3.3 Q 12(a)if the value of α is
(A) NCERT Solutions Class 12 Maths Cahpter 3 Ex 3.3 Q 12(b)(B) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 12(c)(C) π (D) NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 12(d)
Answer:

The correct answer is B.
NCERT Solutions Class 12 Maths Chapter 3 ex 3.3 Q 12(e)

Comparing the corresponding elements of the two matrices, we have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.3 Q 12(f)


NCERT Solutions For Class 12 Maths Chapter 3 Matrices Ex 3.4 Operations On Matrices

Question 1:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 1
Answer:
Let A= NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 1
We know that A = IA
NCERT Solutions Class 12 Maths chapter 3 ex 3.4 Q 1(a)



Question 2:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 2
Answer:
Let A= NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 2

We know that A = IA

NCERT Solutions Class 12 Maths Chapter 3 EX 3.4 Q 2(a)


Question 3:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 3
Answer:
Let A= NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 3
We know that A = IA

NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 3(a)


Question 4:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Chapter 3 Ex 3.4 Q 4
Answer:
Let A= NCERT Solutions Class 12 Chapter 3 Ex 3.4 Q 4

We know that A = IA
NCERT Solutions Class 12 Maths chapter 3 Ex 3.4 Q 4(a)



Question 5:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 5
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 5

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 5(a)



Question 6:

Find the inverse of each of the matrices, if it exists. NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 6
Answer:
Let A= NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 6

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 6(a)



Question 7:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 7
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 7

We know that A = AI
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 7(a)



Question 8:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths chapter 3 Ex 3.4 Q 8
Answer:
Let A= NCERT Solutions Class 12 Maths chapter 3 Ex 3.4 Q 8

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 8(a)



Question 9:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 9
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 9

We know that A = IA
NCERT Solutions Class 12 Maths chapter 3 Ex 3.4 Q 9(a)



Question 10:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 10
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 10

We know that A = AI
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 10(a)



Question 11:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 11
Answer:
Let A= NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 11

We know that A = AI
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 11(a)



Question 12:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 12
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 12

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 12(a)

Now, in the above equation, we can see all the zeros in the second row of the matrix on the L.H.S.

Therefore, A−1 does not exist.


Question 13:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 13
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 13

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 13(a)



Question 14:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 14
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 14

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 14(a)
applying NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 14(b), we have
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 14(c)

Now, in the above equation, we can see all the zeros in the first row of the matrix on the L.H.S.

Therefore, A−1 does not exist.


Question 15:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths chapter 3 ex 3.4 Q 15
Answer:
Let A=NCERT Solutions Class 12 Maths chapter 3 ex 3.4 Q 15

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 15(a)

NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 15(b)



Question 16:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 16
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 16

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 16(a)

Applying R2 → R2 + 3R1 and R3 → R3 − 2R1, we have:
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 16(b)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 16(c)


Question 17:

Find the inverse of each of the matrices, if it exists.
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 17
Answer:
Let A=NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 17

We know that A = IA
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 17(a)
Applying NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 17(b)we have:
NCRET Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 17(c)
NCERT Solutions Class 12 Maths Chapter 3 Ex 3.4 Q 17(d)
NCERT Solutions Class 12 Maths Chapter 3 ex 3.4 Q 17(e)


Question 18:

Matrices A and B will be inverse of each other only if

A. AB = BA

C. AB = 0, BA = I

B. AB = BA = 0

D. AB = BA = I
Answer:

Answer: D

We know that if A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is said to be the inverse of A. In this case, it is clear that A is the inverse of B.

Thus, matrices A and B will be inverses of each other only if AB = BA = I.


NCERT Solutions For Class 12 Maths Chapter 3 Matrices Miscellaneous Solutions

Question 1:
Let NCERT Solutions Class 12 Maths Chapter 3 ms q 1show that NCERT Solutions Class 12 Maths Chapter 3 ms q 1(a)where I is the identity matrix of order 2 and n ∈ N
Answer:
It is given that NCERT Solutions Class 12 Maths Chapter 3 ms q 1
NCERT Solutions Class 12 Maths Chapter 3 ms q 1(b)

We shall prove the result by using the principle of mathematical induction.

For n = 1, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 1(c)

Therefore, the result is true for = 1.

Let the result be true for n = k.

That is,
NCERT Solutions Class 12 Maths Chapter 3 ms q 1(d)

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

Consider
NCERT Solutions Class 12 Maths Chapter 3 ms q 1(e)

From (1), we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 1(f)

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

Thus, by the principle of mathematical induction, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 1(g)


Question 2:
If NCERT Solutions Class 12 Maths Chapter 3 ms q 2, prove that NCERT Solutions Class 12 Maths Chapter 3 ms q 2(a)
Answer:
It is given that NCERT Solutions Class 12 Maths Chapter 3 ms q 2
NCERT Solutions Class 12 Maths Chapter 3 ms q 2(b)

We shall prove the result by using the principle of mathematical induction.

For n = 1, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 2(c)

Therefore, the result is true for n = 1.

Let the result be true for n = k.
That is NCERT Solutions Class 12 Maths Chapter 3 ms q 2(d)

Now, we prove that the result is true for n = k + 1.
NCERT Solutions Class 12 Maths Chapter 3 ms q 2(e)

NCERT Solutions Class 12 Maths Chapter 3 ms q 2(h)

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

Thus by the principle of mathematical induction, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 2(g)


Question 3:
If NCERT Solutions Class 12 Maths Chapter 3 ms q 3then prove NCERT Solutions Class 12 Maths Chapter 3 ms q 3(a)where n is any positive integer
Answer:
It is given that NCERT Solutions Class 12 Maths Chapter 3 ms q 3
NCERT Solutions Class 12 Maths Chapter 3 ms q 3(b)

We shall prove the result by using the principle of mathematical induction.

For n = 1, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 3(c)

Therefore, the result is true for n = 1.

Let the result be true for n = k.

That is,
NCERT Solutions Class 12 Maths Chapter 3 ms q 3(d)
Now, we prove that the result is true for n = k + 1.
NCERT Solutions Class 12 Maths Chapter 3 ms q 3(e)

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

Thus, by the principle of mathematical induction, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 3(f)


Question 4:
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix
Answer:
It is given that A and B are symmetric matrices. Therefore, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 4

Thus, (AB − BA) is a skew-symmetric matrix.


Question 5:
Show that the matrix NCERT Solutions Class 12 Maths Chapter 3 ms q 5(a)is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Answer:
We suppose that A is a symmetric matrix, then NCERT Solutions Class 12 Maths Chapter 3 ms q 5(b)….(1) consider
NCERT Solutions Class 12 Maths Chapter 3 ms q 5(c)
Thus, if A is a symmetric matrix, then NCERT Solutions Class 12 Maths Chapter 3 ms q 5(a)

is a symmetric matrix.

Now, we suppose that A is a skew-symmetric matrix.

Then,
NCERT Solutions Class 12 Maths Chapter 3 ms q 5(d)

Thus, if A is a skew-symmetric matrix, then NCERT Solutions Class 12 Maths Chapter 3 ms q 5(a)

is a skew-symmetric matrix.

Hence, if A is a symmetric or skew-symmetric matrix, then NCERT Solutions Class 12 Maths Chapter 3 ms q 5(a)is a symmetric or skew-symmetric matrix accordingly.


Question 6:
Find the values of xyz if the matrix NCERT Solutions class 12 Maths Chapter 3 ms q 6satisfy the equation NCERT Solutions Class 12 Maths Chapter 3 ms q 6(a)
Answer:
It is given that NCERT Solutions class 12 Maths Chapter 3 ms q 6
NCERT Solutions Class 12 Math Chapter 3 ms q 6(a)
Now,
NCERT Solutions Class 12 Maths Chapter 3 ms q 6(b)

On comparing the corresponding elements, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 6(c)


Question 7:
For what values of NCERT Solutions Class 12 Maths Chapter 3 ms q 7
Answer:
We have
NCERT Solutions Class 12 Maths Chapter 3 ms q 7(a)

∴4 + 4x = 0

⇒ x = −1

Thus, the required value of x is −1.
Question 8:
If NCERT Solutions Class 12 Maths Chapter 3 ms q 8show that NCERT Solutions Class 12 Maths Chapter 3 ms q 8(a)
Answer:
NCERT Solutions Class 12 Maths Chapter 3 ms q 8(b)



Question 9:
Find x, if NCERT Solutions Class 12 Maths Chapter 3 ms q 9
Answer:
We have:
NCERT Solutions Class 12 Math Chapter 3 ms q 9(a)


Question 10:

A manufacturer produces three products xyz which he sells in two markets.

Annual sales are indicated below:

Market

Products

I

10000

2000

18000

II

6000

20000

8000

(a) If unit sale prices of xy and are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50 paise respectively. Find the gross profit.

Answer:

(a) The unit sale prices of xy, and are respectively given as Rs 2.50, Rs 1.50, and Rs 1.00.

Consequently, the total revenue in market I can be represented in the form of a matrix as:
NCERT Solutions Class 12 Maths Chapter 3 ms q 10

The total revenue in market II can be represented in the form of a matrix as:
NCERT Solutions Class 12 Maths Chapter 3 ms q 10(a)

Therefore, the total revenue in market isRs 46000 and the same in market II isRs 53000.

(b) The unit cost prices of xy, and are respectively given as Rs 2.00, Rs 1.00, and 50 paise.

Consequently, the total cost prices of all the products in market I can be represented in the form of a matrix as:
NCERT Solutions Class 12 Maths Chapter 3 ms q 10(b)

Since the total revenue in market isRs 46000, the gross profit in this marketis (Rs 46000 − Rs 31000) Rs 15000.

The total cost prices of all the products in market II can be represented in the form of a matrix as:
NCERT Solutions Class 12 Maths Chapter 3 ms q 10(c)
Since the total revenue in market II isRs: 53000, the gross profit in this market is (Rs:53000 − Rs: 36000) Rs:17000.


Question 11:
Find the matrix X so that NCERT Solutions Class 12 Maths Chapter 3 ms q 11
Answer:
It is given that:
NCERT Solutions Class 12 Maths Chapter 3 ms q 11
The matrix given on the R.H.S. of the equation is a 2 × 3 matrix and the one given on the L.H.S. of the equation is a 2 × 3 matrix. Therefore, X has to be a 2 × 2 matrix.
Now, let NCERT Solutions Class 12 Maths Chapter 3 ms q 11(a)

Therefore, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 11(b)

Equating the corresponding elements of the two matrices, we have:
NCERT Solutions Class 12 Maths chapter 3 ms q 11(c)

Thus, a = 1, b = 2, c = −2, d = 0
Hence, the required matrix X is NCERT Solutions Class 12 Maths Chapter 3 ms q 11(d)


Question 12:
If A and B are square matrices of the same order such that AB =BA, then prove by induction that NCERT Solutions Class 12 Maths Chapter 3 ms q 12Further, prove that NCERT Solutions Class 12 Maths Chapter 3 ms q 12(a)for all n ∈ N
Answer:

A and B are square matrices of the same order such that AB = BA.
NCERT Solutions Class 12 maths Chapter 3 ms q 12(b)

For n = 1, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 12(c)

Therefore, the result is true for n = 1.

Let the result be true for n = k.
NCERT Solutions Class 12 Maths Chapter 3 ms q 12(d)

Now, we prove that the result is true for n = k + 1.
NCERT Solutions Class 12 Maths Chapter 3 ms q 12(e)

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

Thus, by the principle of mathematical induction, we have
NCERT Solutions Class 12 Maths Chapter 3 ms q 12(f)
Now, we prove that NCERT Solutions Class 12 Maths Chapter 3 ms q 12(g)

for all n ∈ N

For n = 1, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 12(h)

Now, we prove that the result is true for n = k + 1.
NCERT Solutions Class 12 Maths Chapter 3 ms q 12(i)

Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have
NCERT Solutions Class 12 Maths Chapter 3 ms q 12(a)for all natural numbers.


Question 13:

Choose the correct answer in the following questions:
If NCERT Solutions class 12 Maths Chapter 3 ms q 13is such that NCERT Solutions Class 12 Maths Chapter 3 ms q 13(a)then
(A) NCERT Solutions Class 12 Maths Chapter 3 ms q 13(b)
(B) NCERT Solutions Class 12 Maths Chapter 3 ms q 13(c)
(C) NCERT Solutions Class 12 Maths Chapter 3 ms q 13(d)
(D) NCERT Solutions Class 12 Maths Chapter 3 ms q 13(e)
Answer:
Answer: C
NCERT Solutions Class 12 Maths Chapter 3 ms q 13(f)

On comparing the corresponding elements, we have:
NCERT Solutions Class 12 Maths Chapter 3 ms q 13(g)


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:
Answer: B
If A is both symmetric and skew-symmetric matrix, then we should have
NCERT Solutions Class 12 Maths Chapter 3 ms q 14

Therefore, A is a zero matrix.


Question 15:
If A is a square matrix such that NCERT Solutions Class 12 Maths Chapter 3 ms q 15then NCERT Solutions Class 12 Maths Chapter 3 ms q 15(a)is equal to
A. A B. I − A C. I D. 3A
Answer:
Answer: C
NCERT Solutions Class 12 Maths Chapter 3 ms q 15(b)