Chapter 7.5, Problem 10P

Using Theorem $\mathrm{7.5.2}$, show that the linear autonomous system

$\frac{dx}{dt}=a_{11}x+a_{12}y$, $\frac{dy}{dt}=a_{21}x+a_{22}y$ $$

does not have a periodic solution (other than $x=0$, $y=0$) if $a_{11}+a_{22}\ne 0$.