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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
Linear algebra
Moderate
Suggested Book
Linear algebra by Friedbarg

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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