# Matrix question regarding determinant, whether singular or not

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.

• hint for counter-example : same pattern than 1+j+j²=0. – Fabrice NEYRET Oct 18 '15 at 13:32

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 $$\omega + \omega^2 +\omega^3 = 0$$