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Trace the trace: TIFR 2019 GS Part A, Problem 10

Understand the problem

Let \(S=\{x \in \Bbb R| x=tr(A)\) for some \(A \in M_4(\Bbb R)\) s.t \(A^2=A\}\)

Then which of the following will describe \(S\):

  1. \(S=\{0,2,4\}\)
  2. \(S=\{0,1/2,1,3/2,2,5/2,3,7/2,4\}\)
  3. \(S=\{0,1,2,3,4\}\)
  4. \(S=[0,4]\)
Source of the problem
TIFR 2019 GS Part A, Problem 10
Topic
Linear algebra
Difficulty Level
Moderate
Suggested Book
Linear algebra by Friedbarg

Start with hints

Do you really need a hint? Try it first!

What can you say about the minimal polynomial of A?
Observe that min poly(A)|x^2-x=x(x-1). Then the minimal poly could be..?
Can you construct the matrix now?
The matrix corresponding to x, x-1 is zero matrix and the identity matrix respectively and what are the matrices corresponding to x(x-1)?  
Can you think about option 3)?

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