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I want to find a real 3x3 matrix $A$ such that $A^{2}+A+I_{3}=0$ with the additional condition that it has at most one zero entry.

How can i compute his entries?

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It doesn't exist. All $3 \times 3$ real matrices have a real eigenvalue, since the characteristic polynomial is a real polynomial of odd degree, and must have a root. Thus, the minimal polynomial must also have a real root, and would have to divide $x^2 + x + 1$. This is not possible, as $x^2 + x + 1$ has no real root.

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