The document is a chapter from a Class XII math textbook on matrices. It contains examples of operations on matrices such as addition, multiplication, and solving systems of equations involving matrices. Some key examples are:
- Finding the order and elements of given matrices
- Determining the possible orders of matrices with a given number of elements
- Computing products of matrices such as AB and BA when the dimensions are compatible
- Solving systems of equations to find unknown variables in matrices set equal to each other
Steps to solve Transportation models by North west corner method are given the presentation. North west corner method is one of the well known methods used to solve the transportation models.
* Evaluate square roots.
* Use the product rule to simplify square roots.
* Use the quotient rule to simplify square roots.
* Add and subtract square roots.
* Rationalize denominators.
* Use rational roots.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
Steps to solve Transportation models by North west corner method are given the presentation. North west corner method is one of the well known methods used to solve the transportation models.
* Evaluate square roots.
* Use the product rule to simplify square roots.
* Use the quotient rule to simplify square roots.
* Add and subtract square roots.
* Rationalize denominators.
* Use rational roots.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
Right foundation at the right stage is the most important factor in the success of any student in exam and in life the APEX IIT / PMT foundation program is aimed at students studying in class XI, who aspire to prepare for engineering / medical entrance in future. The program keeps the school curriculum as base and further upgrades the students’ knowledge to meet the requirements of competitive exams. The program has been design in a way so as to develop orientation of the students as well as to motivate him to excel in competitive exams.
Two Year Classroom Program is the ideal program for students who wish to start early in their quest for a seat at the IITs. This program helps the student not only to excel in IIT-JEE but also in Olympiads & KVPY by building a strong foundation, enhance their IQ & analytical ability and develop parallel thinking processes from a very early stage in their academic career. Students joining this program will have more time to clear their fundamentals and practice extensively for IIT-JEE, their ultimate goal!
Solving ONE’S interval linear assignment problemIJERA Editor
In this paper to introduce a matrix ones interval linear assignment method or MOILA -method for solving wide range of problem. An example using matrix ones interval linear assignment methods and the existing Hungarian method have been solved and compared. Also some of the variations and some special cases in assignment problem and its applications have been discussed ,the proposed method is a systematic procedure, easy to apply and can be utilized for all types of assignment problem with maximize or minimize objective functions
Solving Transportation Problems with Hexagonal Fuzzy Numbers Using Best Candi...IJERA Editor
In this paper, we introduce a Fuzzy Transportation Problem (FTP) in which the values of transportation costs are
represented as hexagonal fuzzy numbers. We use the Best candidate method to solve the FTP. The Centroid
ranking technique is used to obtain the optimal solution.
WHY CHILDREN ABSORBS MORE MICROWAVE RADIATION THAT ADULTS: THE CONSEQUENCES” ...Shahrukh Javed
In today’s world, technologic developments bring social and economic benefits to large sections of society; however, the health consequences of these developments can be difficult to predict and manage. With rapid advancement in technologies and communications, children are increasingly exposed to MWRs at earlier and earlier ages. Microwave radiation causes damaging health effects, especially for children, whose brains are fragile and still developing. MWR penetration is greater relative to developing brain of fetus and child, and they will have a longer penetration of exposure than adults. This concerns the effects of microwave on health because they pervade diverse fields of our lives. Thus, children are at greater risk than adults when exposed to possible human carcinogen[1], computer simulation using MRI scans of children is the only possible way to determine the microwave radiation (MWR) absorbed in specific tissues in children and this radiation is measured in terms of specific absorption rate[2] (SAR). It also includes an assessment of the potential susceptibility of children to MWR and concludes with a recommendation for additional research in present legal limits for exposure to MWR and the development of precautionary policies in the face of scientific uncertainty.
• UNIVERSAL ASYNCHRONOUS RECEIVER TRANSMITTER
• UART is a device that has the capability to both receive and transmit serial data.
• A universal asynchronous receiver-transmitter is a computer hardware device for asynchronous serial communication in which the data format and transmission speeds are configurable.
A heart rate monitor is a personal monitoring device that allows one to measure one's heart rate in real time or record the heart rate for later study. It is largely used by performers of various types of physical exercise.
A universal asynchronous receiver-transmitter is a computer hardware device for asynchronous serial communication in which the data format and transmission speeds are configurable.
As the ages of mankind progressed there was a rapid change in the means of transportation with technology starting from the days of bullockcarts to the present days of planes,drones and space shuttles. Travel speed increased from miles/hr to the speed of light.
The source to destination became so closer to each others. There was an advancement in the field of the air traveling system with the help of airplane. But as the speed increases , the horror of air crash also introduced. Because at a height of 2000m and above if a plane crashes ,it will be a terror for any body. So to take the feed back of the various activities happens in the plane and record them engineers need a mechanism to record such activities .
With any airplane crash, there are many unanswered questions as to what brought the plane down. Investigators turn to the airplane's flight data recorder (FDR) and cockpit voice recorder (CVR), also known as "BLACK BOXES" for answers. In recent times these black boxes helped to know answers for many of issues regarding the crashes and may come to help to solve many cases in the future also.so these black boxes play a vital role in this modern world with the advancement of technology
What is wireless power transmission(WPT)?
Why is WPT?
History of WPT
Types of WPT
Techniques to transfer energy wirelessly
Advantages and disadvantages
Applications
Conclusion
References
The main objective of this current proposal is to make the recharging of the mobile phones independent of their manufacturer and battery make. In this paper a new proposal has been made so as to make the recharging of the mobile phones is done automatically as you talk in your mobile phone!
The microwave signal is transmitted from the transmitter along with the message signal using special kind of antennas called slotted wave guide antenna at a frequency is 2.45 GHz
Why children absorb more microwave radiation than adults the consequencesShahrukh Javed
In today’s world, technologic developments bring social and economic benefits to large sections of society; however, the health consequences of these developments can be difficult to predict and manage.
With rapid advancement in technologies and communications, children are increasingly exposed to MWRs at earlier ages which causes damaging health effects, especially whose brains are fragile and still developing.
Radiation exposure of cell phones & its impact on humansShahrukh Javed
Radiation Exposure Of Cell Phones & Its Impact On Humans.
Electromagnetic Radiation.
Radio Waves Emitted By The Cell Phone.
Cell Phone Base Station.
Antennas On Cell Tower Transmit In The Frequency Range.
Types Of Antennas –Transmission.
Types Of Base Stations-operators.
Possible Health Effects Of ELF-EMF/RFR Exposure Effects & Base Station.
Illness That Have Potential Links To Phone Radiation.
Awareness Among People.
Wealth Vs Health.
The Precautionary Principle.
Conclusion.
Radiation exposure of cell phones & its impact on humans.Shahrukh Javed
RADIATION EXPOSURE OF CELL PHONES & ITS IMPACT ON HUMANS
ELECTROMAGNETIC RADIATION
RADIO WAVES EMITTED BY THE CELL PHONE
CELL PHONE BASE STATION
ANTENNAS ON CELL TOWER TRANSMIT IN THE FREQUENCY RANGE
TYPES OF ANTENNAS –TRANSMISSION.
TYPES OF BASE STATIONS-OPERATORS.
POSSIBLE HEALTH EFFECTS OF ELF-EMF/RFR EXPOSURE EFFECTS & BASE STATION
ILLNESS THAT HAVE POTENTIAL LINKS TO PHONE RADIATION
AWARENESS AMONG PEOPLE
WEALTH VS HEALTH
THE PRECAUTIONARY PRINCIPLE
CONCLUSION
Black Box Corporation, also doing business as Black Box Network Services, is headquartered in the Pittsburgh suburb of Lawrence, Pennsylvania, United States. The company is a provider of communications products.
Automatic image mosaicing an approach based on fftShahrukh Javed
An Image mosaic is a composition generated from a sequence of images which can be obtained by understanding geometric relationships between images. The geometric relations are coordinate transformations that relate the different image coordinate systems. Image mosaic is a technique for creating images which cannot be created by a single frame of the camera, for example satellite imagery. The user gives a series of pictures with overlapping regions and the software returns one large image with all the pictures merged as accurately as possible. When it is impossible to capture the large image in one shot with available equipment, mosaicing process can be used to create an image montage from separately scanned pieces.
This paper goal is to create a script that will stitch two images together to create one larger image. These stitched images, become panoramic views which increase the visual aesthetics of a scene, and are widely sought out for posters, postcards, and other printed materials. This stitching will be performed using point correspondences between the two images.
Delivering Micro-Credentials in Technical and Vocational Education and TrainingAG2 Design
Explore how micro-credentials are transforming Technical and Vocational Education and Training (TVET) with this comprehensive slide deck. Discover what micro-credentials are, their importance in TVET, the advantages they offer, and the insights from industry experts. Additionally, learn about the top software applications available for creating and managing micro-credentials. This presentation also includes valuable resources and a discussion on the future of these specialised certifications.
For more detailed information on delivering micro-credentials in TVET, visit this https://tvettrainer.com/delivering-micro-credentials-in-tvet/
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...NelTorrente
In this research, it concludes that while the readiness of teachers in Caloocan City to implement the MATATAG Curriculum is generally positive, targeted efforts in professional development, resource distribution, support networks, and comprehensive preparation can address the existing gaps and ensure successful curriculum implementation.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Landownership in the Philippines under the Americans-2-pptx.pptx
Chapter 3 matrices
1. Class XII Chapter 3 – Matrices Maths
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Exercise 3.1
Question 1:
In the matrix , write:
(i) The order of the matrix (ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23
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 =
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:
2. Class XII Chapter 3 – Matrices Maths
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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 5:
Construct a 3 × 4 matrix, whose elements are given by
(i) (ii)
Answer
In general, a 3 × 4 matrix is given by
(i)
3. Class XII Chapter 3 – Matrices Maths
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Therefore, the required matrix is
(ii)
4. Class XII Chapter 3 – Matrices Maths
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Therefore, the required matrix is
Question 6:
Find the value of x, y, and z from the following equation:
(i) (ii)
(iii)
Answer
(i)
As the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
5. Class XII Chapter 3 – Matrices Maths
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x = 1, y = 4, and z = 3
(ii)
As the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
x + y = 6, xy = 8, 5 + z = 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 − y = − 2 and x + y = 6, we get x = 2 and y = 4
∴x = 4, y = 2, and z = 0 or x = 2, y = 4, and z = 0
(iii)
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:
y + 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
6. Class XII Chapter 3 – Matrices Maths
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Question 7:
Find the value of a, b, c, and d from the equation:
Answer
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:
a − 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 + d = 13 ⇒ d = 4
∴a = 1, b = 2, c = 3, and d = 4
Question 8:
is a square matrix, if
(A) m < n
(B) m > n
7. Class XII Chapter 3 – Matrices Maths
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(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, is a square matrix, if m = n.
Question 9:
Which of the given values of x and y make the following pair of matrices equal
(A)
(B) Not possible to find
(C)
(D)
Answer
The correct answer is B.
It is given that
Equating the corresponding elements, we get:
8. Class XII Chapter 3 – Matrices Maths
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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
9. Class XII Chapter 3 – Matrices Maths
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Exercise 3.2
Question 1:
Let
Find each of the following
(i) (ii) (iii)
(iv) (v)
Answer
(i)
(ii)
(iii)
(iv) Matrix A has 2 columns. This number is equal to the number of rows in matrix B.
Therefore, AB is defined as:
10. Class XII Chapter 3 – Matrices Maths
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(v) Matrix B has 2 columns. This number is equal to the number of rows in matrix A.
Therefore, BA is defined as:
Question 2:
Compute the following:
(i) (ii)
(iii)
(v)
Answer
(i)
(ii)
11. Class XII Chapter 3 – Matrices Maths
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(iii)
(iv)
Question 3:
Compute the indicated products
(i)
(ii)
12. Class XII Chapter 3 – Matrices Maths
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(iii)
(iv)
(v)
(vi)
Answer
(i)
(ii)
(iii)
13. Class XII Chapter 3 – Matrices Maths
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(iv)
(v)
(vi)
14. Class XII Chapter 3 – Matrices Maths
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Question 4:
If , and , then
compute and . Also, verify that
Answer
15. Class XII Chapter 3 – Matrices Maths
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Question 5:
If and then compute .
Answer
16. Class XII Chapter 3 – Matrices Maths
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Question 6:
Simplify
Answer
Question 7:
Find X and Y, if
(i) and
(ii) and
Answer
(i)
17. Class XII Chapter 3 – Matrices Maths
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Adding equations (1) and (2), we get:
(ii)
Multiplying equation (3) with (2), we get:
18. Class XII Chapter 3 – Matrices Maths
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Multiplying equation (4) with (3), we get:
From (5) and (6), we have:
Now,
19. Class XII Chapter 3 – Matrices Maths
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Question 8:
Find X, if and
Answer
20. Class XII Chapter 3 – Matrices Maths
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Question 9:
Find x and y, if
Answer
Comparing the corresponding elements of these two matrices, we have:
∴x = 3 and y = 3
Question 10:
21. Class XII Chapter 3 – Matrices Maths
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Solve the equation for x, y, z and t if
Answer
Comparing the corresponding elements of these two matrices, we get:
Question 11:
If , find values of x and y.
Answer
22. Class XII Chapter 3 – Matrices Maths
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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 , find the values of x, y, z and
w.
Answer
Comparing the corresponding elements of these two matrices, we get:
23. Class XII Chapter 3 – Matrices Maths
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Question 13:
If , show that .
Answer
24. Class XII Chapter 3 – Matrices Maths
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Question 14:
Show that
(i)
(ii)
Answer
25. Class XII Chapter 3 – Matrices Maths
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(i)
(ii)
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Question 15:
Find if
Answer
We have A2
= A × A
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Question 16:
If , prove that
Answer
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Question 17:
If and , find k so that
Answer
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Comparing the corresponding elements, we have:
Thus, the value of k is 1.
Question 18:
If and I is the identity matrix of order 2, show that
Answer
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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:
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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:
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:
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The bookshop of a particular school 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. 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:
Thus, the bookshop will receive Rs 20160 from the sale of all these books.
Question 21:
Assume X, Y, Z, W and P are matrices of order , and
respectively. The restriction on n, k and p so that 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 n × 3 and 3 × k respectively.
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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 n × 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, k = 3 and p = n are the restrictions on n, k, and p so that will be
defined.
Question 22:
Assume X, Y, Z, W and P are matrices of order , and
respectively. If n = p, then the order of the matrix 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.
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Exercise 3.3
Question 1:
Find the transpose of each of the following matrices:
(i) (ii) (iii)
Answer
(i)
(ii)
(iii)
Question 2:
If and , then verify that
(i)
(ii)
Answer
We have:
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(i)
(ii)
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Question 3:
If and , then verify that
(i)
(ii)
Answer
(i) It is known that
Therefore, we have:
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(ii)
Question 4:
If and , then find
Answer
We know that
Question 5:
For the matrices A and B, verify that (AB)′ = where
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(i)
(ii)
Answer
(i)
(ii)
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Question 6:
If (i) , then verify that
(ii) , then verify that
Answer
(i)
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(ii)
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Question 7:
(i) Show that the matrix is a symmetric matrix
(ii) Show that the matrix is a skew symmetric matrix
Answer
(i) We have:
Hence, A is a symmetric matrix.
(ii) We have:
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Hence, A is a skew-symmetric matrix.
Question 8:
For the matrix , verify that
(i) is a symmetric matrix
(ii) is a skew symmetric matrix
Answer
(i)
Hence, is a symmetric matrix.
(ii)
Hence, is a skew-symmetric matrix.
Question 9:
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Find and , when
Answer
The given matrix is
Question 10:
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
(i)
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(ii)
(iii)
(iv)
Answer
(i)
Thus, is a symmetric matrix.
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Thus, is a skew-symmetric matrix.
Representing A as the sum of P and Q:
(ii)
Thus, is a symmetric matrix.
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Thus, is a skew-symmetric matrix.
Representing A as the sum of P and Q:
(iii)
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Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
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Representing A as the sum of P and Q:
(iv)
Thus, is a symmetric matrix.
Thus, is a skew-symmetric matrix.
Representing A as the sum of P and Q:
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Question 11:
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
The correct answer is A.
A and B are symmetric matrices, therefore, we have:
Thus, (AB − BA) is a skew-symmetric matrix.
Question 12:
If , then , if the value of α is
A. B.
C. π D.
Answer
The correct answer is B.
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Comparing the corresponding elements of the two matrices, we have:
Exercise 3.4
Question 1:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
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Question 2:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
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Question 3:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Question 4:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
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Question 5:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
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Question 6:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
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Question 7:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = AI
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Question 8:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
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Question 9:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Question 10:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = AI
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Question 11:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = AI
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Question 12:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
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.
Answer
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We know that A = IA
Question 14:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Applying , we have:
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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 16:
Find the inverse of each of the matrices, if it exists.
Answer
We know that A = IA
Applying R2 → R2 + 3R1 and R3 → R3 − 2R1, we have:
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Question 17:
Find the inverse of each of the matrices, if it exists.
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Answer
We know that A = IA
Applying , we have:
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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
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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.
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Miscellaneous Solutions
Question 1:
Let , show that , where I is the identity matrix of
order 2 and n ∈ N
Answer
It is given that
We shall prove the result by using the principle of mathematical induction.
For n = 1, we have:
Therefore, the result is true for n = 1.
Let the result be true for n = k.
That is,
Now, we prove that the result is true for n = k + 1.
Consider
From (1), we have:
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Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have:
Question 2:
If , prove that
Answer
It is given that
We shall prove the result by using the principle of mathematical induction.
For n = 1, we have:
Therefore, the result is true for n = 1.
Let the result be true for n = k.
That is
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Now, we prove that the result is true for n = k + 1.
Therefore, the result is true for n = k + 1.
Thus by the principle of mathematical induction, we have:
Question 3:
If , then prove where n is any positive integer
Answer
It is given that
We shall prove the result by using the principle of mathematical induction.
For n = 1, we have:
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Therefore, the result is true for n = 1.
Let the result be true for n = k.
That is,
Now, we prove that the result is true for n = k + 1.
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have:
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:
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Thus, (AB − BA) is a skew-symmetric matrix.
Question 5:
Show that the matrix is symmetric or skew symmetric according as A is symmetric
or skew symmetric.
Answer
We suppose that A is a symmetric matrix, then … (1)
Consider
Thus, if A is a symmetric matrix, then is a symmetric matrix.
Now, we suppose that A is a skew-symmetric matrix.
Then,
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Thus, if A is a skew-symmetric matrix, then is a skew-symmetric matrix.
Hence, if A is a symmetric or skew-symmetric matrix, then is a symmetric or skew-
symmetric matrix accordingly.
Question 6:
Solve system of linear equations, using matrix method.
Answer
The given system of equations can be written in the form of AX = B, where
Thus, A is non-singular. Therefore, its inverse exists.
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Question 7:
For what values of ?
Answer
We have:
∴4 + 4x = 0
⇒ x = −1
Thus, the required value of x is −1.
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Question 8:
If , show that
Answer
Question 9:
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Find x, if
Answer
We have:
Question 10:
A manufacturer produces three products x, y, z 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 x, y and z 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.
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(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 x, y, and z 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:
The total revenue in market II can be represented in the form of a matrix as:
Therefore, the total revenue in market I isRs 46000 and the same in market II isRs
53000.
(b) The unit cost prices of x, y, and z 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:
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Since the total revenue in market I 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:
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
Answer
It is given that:
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
Therefore, we have:
Equating the corresponding elements of the two matrices, we have:
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Thus, a = 1, b = 2, c = −2, d = 0
Hence, the required matrix X is
Question 12:
If A and B are square matrices of the same order such that AB = BA, then prove by
induction that . Further, prove that for all n ∈ N
Answer
A and B are square matrices of the same order such that AB = BA.
For n = 1, we have:
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Therefore, the result is true for n = 1.
Let the result be true for n = k.
Now, we prove that the result is true for n = k + 1.
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have
Now, we prove that for all n ∈ N
For n = 1, we have:
Therefore, the result is true for n = 1.
Let the result be true for n = k.
Now, we prove that the result is true for n = k + 1.
Therefore, the result is true for n = k + 1.
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Thus, by the principle of mathematical induction, we have , for all natural
numbers.
Question 13:
Choose the correct answer in the following questions:
If is such that then
A.
B.
C.
D.
Answer
Answer: C
On comparing the corresponding elements, we have:
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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
Therefore, A is a zero matrix.
Question 15:
If A is square matrix such that then is equal to
A. A B. I − A C. I D. 3A
Answer
Answer: C
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