Systems of Differential Equations · Superchallenge problem

A real $2\times2$ matrix $A$ has $\operatorname{tr}A=-2$ and $\det A>5$, and every solution of $\mathbf x'=A\mathbf x$ satisfies $\mathbf x(t+\pi)=-e^{-\pi}\mathbf x(t)$ for all $t$. Find the smallest possible value of $\det A$.