Select Page

# Understand the problem

The set of nilpotent matrices in $$M_3(R)$$ spans $$M_3(R)$$ considered as an R-vector space (a matrix A is said to be nilpotent if there exists n ∈ N such that $$A^n = 0)$$.
##### Source of the problem
TIFR 2018 Part A, Problem 15
LINEAR ALGEBRA
Medium
##### Suggested Book
Linear Algebra, Hoffman and Kunze

Do you really need a hint? Try it first!

Let’s first try an easier problem first :
• Whether there exists a and b such that I = a N + b M,where N and M are nilpotent matrices and a,b are real numbers ?
• To use the given definition to prove or disprove this we need to take powers and things will become too messy for larger n though here it is 3.[We have to take only $$A^3=0$$ it is a small exercise to verify]
• So let us recapitulate some properties of nilpotent matrices: They are the matrices which has only eigenvalue 0. How to use this?
• We wish to have some linear operator acting on the right side for the RHS is a linear combination and the natural choice is Trace.
• Trace of a nilpotent matrix is 0.
• Now assume that I is a finite linear combination of Nilpotent matrices and then take Trace Operator on both sides.
• Thus LHS has trace n and RHS has trace 0.(Contradiction)
• So the answer is False.
Food for Thought:
• Observe that the order of the matrices is not at all a big issue here!
• Does span of the nilpotent matrices contain all the matrices with trace 0?
• Hint:Try to find all the 2×2 nilpotent matrices and check the above statement out!

# 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

## Application of eigenvalue in degree 3 polynomial: ISI MMA 2018 Question 14

This is a cute and interesting problem based on application of eigen values in 3 degree polynomial .Here we are finding the determinant value .

## Order of rings: TIFR GS 2018 Part B Problem 12

This problem is a cute and simple application on the ring theory in the abstract algebra section. It appeared in TIFR GS 2018.

## Last three digit of the last year: TIFR GS 2018 Part B Problem 9

This problem is a cute and simple application on the number theory in classical algebra portion. It appeared in TIFR GS 2018.

## Group in graphs or graphs in groups ;): TIFR GS 2018 Part A Problem 24

This problem is a cute and simple application on the graphs in groups in the abstract algebra section. It appeared in TIFR GS 2018.

## Problems on quadratic roots: ISI MMA 2018 Question 9

This problem is a cute and simple application on the problems on quadratic roots in classical algebra,. It appeared in TIFR GS 2018.

## Are juniors countable if seniors are?: TIFR GS 2018 Part A Problem 21

This problem is a cute and simple application on the order of a countable groups in the abstract algebra section. It appeared in TIFR GS 2018.

## Coloring problems: ISI MMA 2018 Question 10

This problem is a cute and simple application of the rule of product or multiplication principle in combinatorics,. It appeared in TIFR GS 2018.

## Diagonilazibility in triangular matrix: TIFR GS 2018 Part A Problem 20

This problem is a cute and simple application on the diagonilazibility in triangular matrix in the abstract algebra section. It appeared in TIFR GS 2018.

## Multiplicative group from fields: TIFR GS 2018 Part A Problem 17

This problem is a cute and simple application on the Multiplicative group from fields in the abstract algebra section. It appeared in TIFR GS 2018.

## Group with Quotient : TIFR GS 2018 Part A Problem 16

This problem is a cute and simple application on Group theory in the abstract algebra section. It appeared in TIFR GS 2018.