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Let $A$ be a nonzero square matrix such that $$A + A^2 + A^3 =0$$

Must $A$ be singular? If your answer is affirmative, give a proof, give a counterexample otherwise.

I have $(\mathop{\text{det}}A)(\mathop{\text{det}}(1+A+A^2)=0$ so that $\mathop{\text{det}}A=0$ or $\mathop{\text{det}}(1+A+A^2)=0$ but I do not know how to continue.

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  • $\begingroup$ hint for counter-example : same pattern than 1+j+j²=0. $\endgroup$ – Fabrice NEYRET Oct 18 '15 at 13:32
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Consider the square matrix $$A=e^{\frac {2\pi i}{3}}1$$, where $1$ is the identity matrix. Since $\omega =e^{\frac {2\pi i}{3}}$ is the primitive cube root of unity, it satisfies $$ \omega + \omega^2 +\omega^3 = 0 $$

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  • $\begingroup$ For homeworks we should give hints, not the solution. $\endgroup$ – Fabrice NEYRET Oct 18 '15 at 16:47
  • $\begingroup$ Well, I was not aware that it might be a homework problem. Now that I look at it again it seems like one. $\endgroup$ – BigbearZzz Oct 19 '15 at 6:04

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