HELP Consider the following phase portrait correspondent to

HELP!!!

Consider the following phase portrait correspondent to a linear system of first order. Choose the one statement below that is true: A. Both eigenvalues of the coefficient matrix are positive. B. Both eigenvalues of the coefficient matrix are real values. C. Both eigenvalues of the coefficient matrix are pure imaginary. D. The coefficient matrix is defective. E. One eigenvalue is zero.

Solution

So we see that the the phase portrait consists of ellipses so there are no real values else solution would either spiral in or spiral out to infinity. Moving in ellipse implies purely imaginary and distinct eigenvalues

So, C. is the correct option.