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$A$ is a 3x3 matrix over $\mathbb{C}$ that's not diagonalizable with trace 3 and determinant 1. Instructions: Find all possible characteristic and minimal polynomials.

Characteristic polynomials: The trace and determinant correspond to the constant term and degree $n-1$ term so that $c_A(x)=x^3-3x^2+\square x-1$. I am stuck here. Does the diagonalizable condition give constraints on $\square$ and the choices of minimal polynomials?

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Hint

Since $A$ is not diagonalizable, the characteristic polynomial must have a repeated root.

Equivalently, the characteristic polynomial and its derivative have a common root.

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