Select Page

# Understand the problem

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.
##### Source of the problem
TATA INSTITUTE OF FUNDAMENTAL RESEARCH GS-2018 (Mathematics)
Linear Algebra
Medium
##### Suggested Book
Linear Algebra Hoffman and Kunze

Do you really need a hint? Try it first!

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
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
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
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.

# 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.

# Similar Problems

## Problem on Inequality | ISI – MSQMS – B, 2018 | Problem 2a

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

## Data, Determinant and Simplex

This problem is a beautiful problem connecting linear algebra, geometry and data. Go ahead and dwelve into the glorious connection.

## Problem on Integral Inequality | ISI – MSQMS – B, 2015

Try this problem from ISI MSQMS 2015 which involves the concept of Integral Inequality and real analysis. You can use the sequential hints provided to solve the problem.

## Inequality Problem From ISI – MSQMS – B, 2017 | Problem 3a

Try this problem from ISI MSQMS 2017 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

## Problem on Natural Numbers | TIFR B 2010 | Problem 4

Try this problem of TIFR GS-2010 using your concepts of number theory and congruence based on natural numbers. You may use the sequential hints provided.

## Definite Integral Problem | ISI 2018 | MSQMS- A | Problem 22

Try this problem from ISI-MSQMS 2018 which involves the concept of Real numbers, sequence and series and Definite integral. You can use the sequential hints

## Inequality Problem | ISI – MSQMS 2018 | Part B | Problem 4

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality and Combinatorics. You can use the sequential hints provided.

## Problem on Inequality | ISI – MSQMS – B, 2018 | Problem 4b

Try this problem from ISI MSQMS 2018 which involves the concept of Inequality. You can use the sequential hints provided to solve the problem.

## Positive Integers Problem | TIFR B 201O | Problem 12

Try this problem of TIFR GS-2010 using your concepts of number theory based on Positive Integers. You may use the sequential hints provided.

## CYCLIC GROUP Problem | TIFR 201O | PART A | PROBLEM 1

Try this problem from TIFR GS-2010 which involves the concept of cyclic group. You can use the sequential hints provided to solve the problem.