# Find a real 3x3 matrix A that satisfies a quadratic equation

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?

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.