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
Difficulty Level
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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