Let $A\in \mathbb{R}^{n\times n}$ be a matrix that has exactly one real eigenvalue, the $-1$. Find the maximal interval of existence of solution $X:\mathbb{R}\rightarrow \mathbb{R}^{n\times n}$ of $$X'(t)=X^2(t) \\ X(0)=A$$
That $-1$ is an eigenvalue of $A$ we have that there is a vector $v$ such that $-A=-v$, right?
If we consider the problem in an interval of the form $(-a,a)$ then we know that there is a solution, or not? So we have to expand this interval as possible, or what do we have to do to find the maximal interval?