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# 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!

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