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Find a matrix whose characteristic polynomial is $\newcommand{\l}{\lambda}\l^6-5\l^5+3\l^3+4\l^2-\l+5.$

Given this characteristic polynomial, I tried factorizing it to obtain its roots, then work from there to find an original matrix $A$. However, here the roots seem either irrational or complex by rational root theorem, which makes this approach impossible. Are there other characteristics of the eigenvalues that I can use here to find an original matrix $A$?

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1 Answer 1

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Hint: Find the companion matrix of the polynomial.

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