Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2009 / Differential equations / 8 / i / Solution

The principal matrix is $\left(\begin{array}{cc}a& b\\ b& c\end{array}\right)$. The equation is elliptic at a point when $\boxed{b^2-ac<0}$, equivalently the principal form has eigenvalues of the same nonzero sign and no real characteristic directions.