# 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!

# Watch the video

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