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College Mathematics

Spanning matrix space by niltopent matrices: TIFR 2018 Part A, Problem 15

This problem is a cute and simple application of spanning matrix space by niltopent matrices in the linear algebra section. It appeared in TIFR GS 2018.

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
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!

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!

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