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 \(S\):

- \(S=\{0,2,4\}\)
- \(S=\{0,1/2,1,3/2,2,5/2,3,7/2,4\}\)
- \(S=\{0,1,2,3,4\}\)
- \(S=[0,4]\)

TIFR 2019 GS Part A, Problem 10

Linear algebra

Moderate

Linear algebra by Friedbarg

Do you really need a hint? Try it first!

What can you say about the minimal polynomial of ?

Observe that min poly. Then the minimal poly could be..?

Can you construct the matrix now?

The matrix corresponding to is zero matrix and the identity matrix respectively and what are the matrices corresponding to ?

Can you think about option 3)?

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