# IIT JAM Mathematics 2022: All you need to know to prepare for it

IIT JAM Mathematics (MA) is considered one of the most sought-after master’s level competitive exams after BSc./B.Tech. Students can get direct admission into IITs and into IISC (upon clearing the interview). The IISER’s also take IIT JAM rank into account so all in all, it is a pretty important entrance if you are interested in higher mathematics.

## IIT JAM Mathematics (MA) 2022: Important Details

### Eligibility for IIT JAM MA 2022

To apply for admission to the Mathematics program, a candidate is required to qualify in the Mathematics Test Paper and also satisfy the Minimum Educational Qualifications (MEQs) and Eligibility Requirements (ERs) of the Academic Programme.

The Eligibility Requirements for IIT JAM 2022 is given below:

• All candidates admitted through JAM should have a Bachelor’s degree.
• In the qualifying degree, the aggregate marks or CGPA/CPI without rounding-off (taking into account all subjects, including languages and subsidiaries, all years combined) should be at least 55% or 5.5 out of 10 for General/OBC (NCL)/EWS category candidates, and 50% or 5.0 out of 10 for SC/ST and PwD category candidates.
• Foreign nationals with Indian degree are eligible to apply, subject to the admitting institute’s policy.

Check out the official website of IIT JAM for the Application procedure, detailed Eligibility Requirements (ER) and Minimum Educational Qualification (MEQ), application fee, etc: https://jam.iitr.ac.in/

### Duration of the course

The duration of the M.Sc course for Mathematics course is 2 years.

### Important Dates:

• Online Application Opens: 30th August 2021
• Last Date for Online Registration: 11th October 2021
• Availability of JAM Admit Cards: 4th January 2022
• Date of Examination: 13th February 2022
• Announcement of the Results: 22nd March 2022

### Admitting Institutes for IIT JAM Mathematics

Candidates can secure admission to the following colleges:

• Indian Institure of Science (IISc), Bangalore
• IIT Bhilai (IITBH)
• IIT Bhubaneswar (IITBBS)
• IIT Bombay (IITB)
• IIT Delhi (IITD)
• IIT Gandhinagar (IITGN)
• IIT Guwahati (IITG)
• IIT Indore (IITI)
• IIT Jodhpur (IITJ)
• IIT Kanpur (IITK)
• IIT Kharagpur (IITKgp)
• IIT Mandi (IITMandi)
• IIT Patna (IITP)
• IIT Roorkee (IITR)
• IIT Ropar (IITRPR)
• IIT Tirupati (IITTP)
• IIT (BHU) Varanasi (IITBHU)

Cheenta is initiating an Open for All Math Camp for College Students and High School Passouts who really love Mathematics and want to pursue it for higher studies.

To get free resources for your IIT JAM Mathematics 2022 Preparation, Register here.

### Exam Pattern for IIT JAM Mathematics 2022

The JAM 2022 Examination for all the seven Test Papers will be carried out as ONLINE Computer Based Test (CBT) where the candidates will be shown the questions in a random sequence on a computer screen. This is how the Exam pattern for IIT JAM Mathematics 2020 will look like:

The time duration of the Exam is 3 hours in total.

Marking scheme:

In all sections, questions not attempted will result in zero marks. In Section-A (MCQ), the wrong answer will result in negative marks. For each wrong answer to one-mark questions, the one-third mark will be deducted and similarly, for each wrong answer to two-mark questions, a two-third mark will be deducted. In Section-B (MSQ), there are no negative and no partial marking provisions. There is no negative marking in Section-C (NAT) as well.

### Syllabus

The syllabus for IIT JAM MA includes:

• Real Analysis
• Multivariable Calculus and Differential Equations
• Linear Agebra and Algebra

The Open for All Math Camp gives an opportunity to students to learn live from the experts for free.

## How to prepare for IIT JAM Mathematics Entrance 2021

### Books for IIT JAM Mathematics (MA) Entrance

#### Books for Linear Algebra

• Chapter 0 of Serge Lang’s Introduction Linear Algebra
• Introduction to Linear Algebra by Gilbert Strang
• Linear Algebra: A geometric Approach by S Kumaresan
• Linear Algebra by Hoffman & Kunze
• Algebra by Artin
• Topics in Algebra by Herstein
• Linear Algebra Done Right by Axler

#### Books for Abstract Algebra

• Contemporary Abstract Algebra by Gallian
• SAGE Math (computation software)
• Abstract Algebra by Dummit and Foote

#### Books for Real Analysis

• Introduction to Real Analysis by Bartle Sherbet
• Problems in Real Analysis by Kaczor
• A basic course in Real Analysis by Kumar & Kumaresan
• Analysis 1 & 2 by Terence Tao
• Principles of Mathematical Analysis by Rudin

#### Books for Vector Calculus and Differential Equation

• Calculus, Early Transcendentals by James Stewart
• Basic Multivariable Calculus by Marsden, Tromba, and Weinstein.
• Calculus Vol 1 and 2 by Apostle
• Calculus on Manifolds by Spivak

### Preparation Tips for IIT JAM MA 2022

Go through the syllabus:-

Learning about the syllabus is an essential part of preparation. This helps to have a better understanding of the preparation strategy and what topics to focus on. It saves you time as well.

Work on your concept and solve different questions:-

Try to clear your concepts on each topic and practice questions based on that topic from different books. This way, you will be able to understand your understanding of the topic.

Solve Previous Papers:-

This will help you know the exact difficulty level of the exam and prepare accordingly. Try to solve them in a time-bound manner. Check out some of the past year's problems available here.

Take Mock Tests:-

Take Mock Tests to check your ability of problem-solving. If it is an online test, then it can help you know your preparation status in the competition. You will be able to know where you stand and what should be improved.

You should also track your progress. If something unimportant is stealing your time, recognize it and try to avoid it. Keep yourself surrounded by people who encourage you. Discuss your dreams and goals with them.

Take proper rest and believe in yourself:-

It is the biggest myth that studying for long hours brings productivity to your work. There should be a routine with a proper work-rest balance and then you should ritually follow it. Also having low self-esteem can ruin your performance in exams.

150+ Students are already part of the Open for All Math Camp

### Prepare for IIT JAM Mathematics (MA) with Cheenta

Cheenta offers a full-year program for college students, willing to crack the IIT JAM Mathematics 2022 Entrance Exam. The program includes classes on Pre-College Mathematics, Real Analysis, Linear Algebra, Abstract Algebra & Vector Calculus. The classes for the full-year program includes:

1. Two Group classes per week (conceptual and problem solving)
3. Regular Quizzes and Assignments

Get a Trial Class now!

# What is TIFR and how to prepare for it?

Tata Institute of Fundamental Research, TIFR is the foremost institution for advanced research in foundational sciences based in Mumbai, Maharashtra, India. The institute offers a master's course, an integrated M.Sc and Ph.D. course and a Ph.D. degree in different science fields. One can get admission into this institute by clearing the TIFR Entrance Exam. It is a perfect place for pioneering scientists to commence their careers.

## TIFR Entrance for Integrated M.Sc-Ph.D. Programme in Mathematics

### Eligibility

The eligibility criterion to be considered for admission to the Integrated Ph.D. programme is a Bachelor's degree in any of Mathematics, Statistics, Science or Technology (B.A. / B.Sc. / B.Math. / B.Stat. / B.E. / B.Tech.) from any recognized university.

### Duration of the course

The duration of integrated courses is 6 years.

### Application Procedure

Candidates can fill the application form online through the university website. After filling the form, candidates can pay the application fee online through internet banking or Debit/Credit Card.

Generally, the admission process takes place in the month of December.

Candidates can secure the admission in the following colleges:

• TIFR, Mumbai
• CAM, Bengaluru
• HBCSE, Mumbai
• ICTS, Bengaluru
• NCRA, Pune
• NCBS, Bengaluru

### Selection Procedure

The selection of the candidates will be done on the basis of the Graduate School (GS) exam. A list of selected candidates will be published on the official website, on the basis of which, interview round will take place. This interview will be conducted by the university.

### TIFR GS Entrance Exam Pattern

The time duration of the Exam is 3 hours in total.

This is the latest exam pattern followed in 2020 paper and patterns may change.

### Syllabus

The syllabus includes:

• Algebra
• Real Analysis
• Geometry / Topology
• General

## How to prepare for TIFR Entrance

### Books for TIFR Entrance

#### Books for Linear Algebra

• Chapter 0 of Serge Lang’s Introduction Linear Algebra
• Introduction to Linear Algebra by Gilbert Strang
• Linear Algebra: A geometric Approach by S Kumaresan
• Linear Algebra by Hoffman & Kunze
• Algebra by Artin
• Linear Algebra Done Right by Axler

#### Books for Abstract Algebra

• Contemporary Abstract Algebra by Gallian
• SAGE Math (computation software)
• Abstract Algebra by Dummit and Foote
• Topics in Algebra by Herstein

#### Books for Real Analysis

• Introduction to Real Analysis by Bartle Sherbet
• Problems in Real Analysis by Kaczor
• A basic course in Real Analysis by Kumar & Kumaresan
• Analysis 1 & 2 by Terence Tao
• Principles of Mathematical Analysis by Rudin

#### Books for Topology

• Topology of Metric Spaces by Kumaresan
• Topology by Munkres

#### Books for Vector Calculus and Differential Equation

• Calculus, Early Transcendentals by James Stewart
• Basic Multivariable Calculus by Marsden, Tromba, and Weinstein.
• Calculus Vol 1 and 2 by Apostle

### Preparation Tips

Go through the syllabus:-

Learning about the syllabus is an essential part of preparation. This helps to have a better understanding of the preparation strategy and what topics to focus on. It saves you time as well.

Work on your concept and solve different questions:-

Try to clear your concepts on each topic and practice questions based on that topic from different books. This way, you will be able to understand your understanding of the topic.

Solve Previous Papers:-

This will help you know the exact difficulty level of the exam and prepare accordingly. Try to solve them in a time-bound manner.

Take Mock Tests:-

Take Mock Tests to check your ability of problem-solving. If it is an online test, then it can help you know your preparation status in the competition. You will be able to know where you stand and what should be improved.

You should also track your progress. If something unimportant is stealing your time, recognise it and try to avoid it. Keep yourself surrounded by people who encourage you. Discuss your dreams and goals with them.

Take proper rest and believe in yourself:-

It is the biggest myth that studying for long hours brings productivity at your work. There should be a routine with a proper work-rest balance and then you should ritually follow it. Also having low self-esteem can ruin your performance in exams.

All the best!

### Prepare for TIFR Mathematics with Cheenta

Cheenta offers a full year program for college students, willing to crack the TIFR Entrance. The Program includes classes on Pre-College Mathematics, Real Analysis, Linear Algebra, Abstract Algebra & Vector Calculus. The classes for the full year program includes:

1. Two Group classes per week (conceptual and problem solving)
3. Regular Quizzes and Assignments

Get a Trial Class now!

# Understand the problem

The number of rings of order 4, up to isomorphism, is:
(a) 1
(b) 2
(c) 3
(d) 4.

Hint 1:

First, ask yourself how many groups are there of order 4.
the answer is simple => Z/4Z and Klein’s four group (K).

Hint 2 :
So intuitively there should be two rings with 4 elements.
Okay, I will give you two rings or same order=> (R,+,.), (R,+,*) . see that order of the rings are same but I have changed the multiplication, and

I define a*b=0 for all a,b in R. [(R,+,*) is called zero ring]

Hint 3:
Question: Prove that (R,+,.) and (R,+,*) are not isomorphic! (easy)
So, we had (Z/4Z,+,.) and (K,+,.) as our answers. But if you change the multiplication to “*” then there will be 4 different rings *upto isomorphism* right?
Hence the answer is 4.
Hint 4:
Bonus Problem:Prove that there are only two non-ismorphic p-rings(ring with p elements) upto isomorphism.

# Connected Program at Cheenta

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]What are the last 3 digits of $$2^{2017}$$?
(a) 072
(b) 472
(c) 512

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.2.1"]Pretty convenient problem for number theory lovers. I’m going to give some insight from Group Theory and solve with basic congruence techniques.
1. Now, an obvious fact is $$2^{2017}$$ ≅ $$0(mod8)$$
2. so my idea is to find modul0 1000. so where does group theory lend you a hand?
3. see that 2 is a generator of (Z/125Z)* (why?)
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.2.2"]
So we get $$2^{2017}$$ ≅ $$2^{2017}(mod125)$$ ≅ $$2^{17}$$ ≅ $$72 (mod 125)$$
4. Now, what is the most demanding step after this?
combining 1 and 3 we get $$2^{2017}$$ ≅ $$072(mod 1000)$$ [as $$125|2^{2017}-72, 8|2^{2017}-72$$=> third digit is 0]
[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.2.3"]
Bonus Problems:
Q. find last two digits of 2^2016 like this process.
SOME NUMBER THEORIC PROBLEMS:
Q. Prove Wilson's theorem using basic group theory
Q. Prove Wilson's theorem by using Sylows theorems.

# Graphs in groups or Groups in graphs : TIFR GS 2018

## Understand the problem

Let G be a finite group and g ∈ G an element of even order. Then we
can colour the elements of G with two colours in such a way that x and
gx have different colours for each x ∈ G.

Do you really need a hint? Try it first!

Hint 1
One needs to know the basics of Graph Theory to understand the solution.
• As noted Colouring is a fundamental topic in Graph Theory,so we need to convert the problem into a graph theory problem.
• Consider the elements of the group G as the vertices and consider edges between two elements say x and y if x=gy or $$x=(g^{-1})x$$. Call graph G*.
• Check that x---y---z (x is adjacent to y and y is adjacent to z) iff $$y = (g^{-1})x , z = (g^{-1})y = (g^{-2})x$$ or $$y = x , z = (g)y = (g^2)x$$.We will assume left multiplication by g ( the proof for $$g^{-1}$$ is exactly the same.)
Hint 2
• Now observe that we need to answer whether this graph G* is 2-colourable.
• There is an elementary result in graph theory characterizing the 2-colourable graphs.
Theorem 1 : A graph is 2-colourable iff it is bipartite.
Theorem 2: A graph is bipartite iff it has no odd-cycle.
Hint 3 :
• Thus Theorem 1 and Theorem 2 ⇒ G* is 2 -colourable iff it has no odd cycle.
• Now what does odd cycle mean in here in terms of group.
• A path of odd length means that $$x---gx---(g^2)x---...---(g^k)x$$ in odd number of steps i.e. k is odd.
Hint 4 :
• A cycle of odd length means that $$(g^k)x=x ⇒ g^k=1$$.
• We are given that g is even ordered so it can only happen if k is even.
• Hence an odd cycle cannot exist and we can colour G* with 2 colours.

## Connected Program at Cheenta

College Mathematics Program the higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Understand the problem

A countable group can have only countably many distinct subgroups.

Hint 1
This is a really artistic problem.
Pre-Solution Thoughts:
Lemma: A group is finite iff the number of subgroups of a group is finite.(Check!)
The lemma is a way to understand that an infinite group will have infinite number of subgroups.

Hint 2
• We are considering here set of all subsets of a countable group G, which is a subset of the Power Set of G.
• Given G is countable the Power Set of G is uncountable.
• Now we know that $$2^d$$, where d is the set of natural numbers denotes the cardinality of the Power Set of G which is an uncountable set as it is bijective to the Real Numbers [Here d is not finite by the way, d is countably infinite].
• So it is kinda intuitive that it may be uncountable.
• First I took the group (Z,+), but we all know that the subgroups of  Z are nZ only. This doesn’t solve our purpose.
• So naturally the next choice was the group (Q.+) whose subgroup is (Z,+).
• While understanding the subgroups of (Q.+), the question is solved.
• Hint 3
• We need to understand the subgroups of (Q.+).
• Consider any rational number q and the subgroup qZ of (Q.+) generated by q.
• What if we take two rational numbers?
• For simplicity check the subgroup generated by {1/2 , 1/3}.
• Prove that the subgroup generated by {1/2 , 1/3} is (1/2.3)Z.{Observe that co-prime property of 2 and 3 is playing an important role}.
• Hint 4
• Now what if we take three mutually coprime natural numbers say a,b,c and see the subgroup generated by { 1/a , 1/b ,1/c}. In fact for simplicity take a,b,c to be primes.
• Observe that it is of the form (1/a.b.c)Z.
• Hence for every finite subset of primes, we generate a distinct subgroup of (Q,+).
• Hence the total number of subgroups contained all the subsets of primes, which has a bijection with the Real Numbers as mentioned above.
• So the answer is False.
• Exercise
• Find all the subgroups of (Q,+) with proof.

# Connected Program at Cheenta

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Understand the problem

Let G be a finite group with a normal subgroup H such that G/H has
order 7. Then $$G \cong$$ H × G/H.

Hint 1
This is also an interesting question. First of all we need to understand something in general.
If G is a finite group and H Δ G. So Consider the quotient group G/H.
Observe the following!
• Lemma: If G ≈ H x G/H , then G/H is isomorphic to a normal subgroup of G. [Consider the projection homomorphism of G to (H,1) which contains G/H as the kernel.]
• But in general G/H is not even a subgroup of G.
We will illustrate this by giving a simple example.
Hint 2
• Naturally we took the group (Z,+) and we know all the subgroups of Z are nZ ,which are normal subgroups as Z is an abelian group.
• Consider the quotient group Z/nZ.We know that this is not even isomorphic to a subgroup of Z.
• Hence comes our counter-example.
• G=Z, H=7Z. G/H =Z/7Z. But G is not isomorphic to 7Z x Z/7Z as Z/7Z is not isomorphic to a subgroup of Z.
• Hence the answer is False.
Hint 3
1. But we will give an example where the given statement is also False.
• Consider the Dihedral group on n elements $$D_n$$ as a subgroup of O(2) [The orthogonal group in $R^2$.] There is a homomorphism (Determinant) from O(2) → {-1,1},whose kernel is SO(2).
• Hence consider the homomorphism from $$D_n$$ → {-1,1} formed by the composition of inclusion homomorphism and the determinant homomorphism.
• Observe that the Kernel of the above defined homomorphism is the Rotation Group of angle 2 π /n and the Quotient Group is the Reflection Group around a specific line(?)[Which is essentially Z/2Z.]
• But observe that Dn is not isomorphic to Rotation Group of angle 2 π /n x Z/2Z .[As there is an interaction between rotation and reflection. $$ref.rot.ref=rot^{-1}$$ .]
Hint 4
1. Prove that the finite subgroups of the group of rigid body motion are only
• Rotation Group of Angle 2 π /n for all n in N.
• Dihedral Group $$D_n$$

# Connected Program at Cheenta

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]The permutation group $$S_{10}$$ has an element of order 30.[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]TIFR 2018 Part A Problem 23[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="off"]Group Theory[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Easy[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Dummit and Foote[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.2.1"]
Consider S={1,2,...,10}.$$S_{10}$$ be the permutation group on S.
What will you do if one asked for a subgroup of order 3!=6?
• We all would have taken the subgroup of $$S_3$$ embedded in $$S_{10}$$ right? Call the subgroup H taking identity transformation on {4,5,6,..10} and embedding in {1,2,3}.
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.2.2"]
What do you do if one asked for a subgroup of order 5?
• We will try to form a cyclic subgroup of order 5. We need to find a generator. Can you see it?
• Keeping {1,2,.,5} same and taking 5 elements {6,7,..,10} and observe the permutation $$i\mapsto i+1$$ for i=6,7,8,9 and $$10\mapsto 7$$.Take the subgroup generated by this element.Observe this is a cyclic subgroup of order 5.Call this subgroup K.
• Now any idea how to combine this?
[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.2.3"]
• Observe that HK is a set of 30 elements. Does it seem HK is a subgroup?
• Lemma: H and K are two subgroups of G. HK is a subgroup of G iff KH=HK.
• Using the lemma prove that HK is really a subgroup of $$S_{10}$$ of order 30.
• Observe that the selection of disjoint elements of H and K is the main reason behind this!
• Hence the answer is True.

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" hover_enabled="0" _i="1" _address="0.0.0.1"]Let A, B ∈ $$M_n(\Bbb R)$$ be such that A + B = AB. Then AB = BA. [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]TIFR 2018 Part A Problem 19[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="off"]Linear Algebra[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Medium[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Linear Algebra, Hoffman and Kunze[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"]

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.2.1"]
We need to play around with symbols.
Try to make (AB-BA) containing equation so that we can get the idea how to show it 0.This is a symmetric operation so remember if you use only B to get something you can also use A to get that so we have to take help of the symmetry.
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.2.2"]
• Playing with the equation and multiplying the equation once left and once right by B we get A.B+B.B=A.B.B and B.A + B.B = B.A.B
• Now subtract it to get AB-BA term,which gives (AB-BA)=(AB-BA)B => (AB-BA)(I-B)=0
• Similarly use A in place of B to get (I-A)(AB-BA)=0
[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.2.3"]
• Now if somehow we can show (I-A) or (I-B) is invertible then we are done.
• So we go back to the actual equation. Remember the identity (1-a)(1-b)=1-a-b+ab.
• Observe that (I-A)(I-B)=I follows from the actual equation.Hence the result follows as (I-A) and (I-B) are invertible and are inverses of each other.
• The answer is therefore True.
[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27" hover_enabled="0" _i="4" _address="0.1.0.2.4"]
U+T are linear operators on R^n and U+T=UT.Then Img(T)=Img(U) and Ker(T)=Ker(U)
• Prove that Ker(T) is a subspace of Ker(U).
• Take adjoint of the equation and observe that Ker(U*) is a subspace of Ker(T*).
• Now we know that Ker(U*) is orthogonal to Img(U) and same goes for T.
• Hence conclude by that Img(T) is a subspace of Img(T).
• Therefore prove that Img(T)=Img(U) and Ker(T)=Ker(U) by Rank Nullity Theorem.

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" hover_enabled="0" _i="1" _address="0.0.0.1"]True or False: Let A, B, C ∈ M3($$\mathbb{R}$$) be such that A commutes with B, B commutes with C and B is not a scalar matrix. Then A commutes with C. [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" _i="0" _address="0.1.0.0.0" hover_enabled="0"]TATA INSTITUTE OF FUNDAMENTAL RESEARCH GS-2018 (Mathematics) [/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" _i="1" _address="0.1.0.0.1" open="off" hover_enabled="0"]Linear Algebra [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" _i="2" _address="0.1.0.0.2" open="off" hover_enabled="0"]Medium [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" _i="3" _address="0.1.0.0.3" open="off" hover_enabled="0"]Linear Algebra Hoffman and Kunze [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"]

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" _i="1" _address="0.1.0.2.1" hover_enabled="0"]First we need to get some idea whether or not this may be true or false  . As a result we need to make some calculations. The first step to approach is always to build the expression of AC - CA & then see whether it is zero or not [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" _i="2" _address="0.1.0.2.2" hover_enabled="0"]Given AB = BA   &  BC = CB . Prove the following ! B(AC) = AC(B) & B(CA) = (CA)B and then subtract to get ( AC - CA )B =  B(AC -CA) Now this is not obvious if DB = BD & B being non- scalar matrix then D = 0 is too strong statement to be true  So this gives us idea  that it maybe false . Now to prove it false we need to construct a counter example [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" _i="3" _address="0.1.0.2.3" hover_enabled="0"]We can take approach using beautiful fact of matrices that they are transformation of spaces . Now given they are transformation of spaces and this is sort of abelian character showing up , we seek help from Groups     [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27" _i="4" _address="0.1.0.2.4" hover_enabled="0"]You know why B is restricted to be non scalar because they form the centre of the $$GL_n$$(F) So we approach it in the following way If we can find a group of 3 x 3 matrices with non trivial centre . If we search for centre of groups then the only example available to us is Heisenberg Group In mathematics, the Heisenberg group    , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers(resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to n-dimensional systems, and most generally, to any symplectic vector space. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]