Understand the problem
Source of the problem
Start with hints
- 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.
- 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
Connected Program at Cheenta
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.