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College Mathematics

Commutative does not commute in matrices: TIFR 2018 Part A, Problem 11

This problem is a cute and simple approach using beautiful fact of matrices transformation in the linear algebra section. a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers or the ring of integers .It appeared in TIFR GS 2018.

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)
Topic
Linear Algebra
Difficulty Level
Medium
Suggested Book
Linear Algebra Hoffman and Kunze

Start with hints

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.

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