 Solution

Freeze the coefficients at a point. Characteristic normals $(\tau ,\xi )\ne (0,0)$ satisfy the homogeneous quadratic matrix pencil equation $\det (\tau A+\xi B)=0$. In the elementary characteristic classification, two distinct real characteristic directions give a hyperbolic system, no real directions an elliptic system, and a repeated real direction the parabolic case. An identically zero determinant polynomial is a further degeneracy, not covered by that nondegenerate classification.

When $A$ is invertible, write $C=A^{-1}B$. The finite characteristic slopes solve $\det (B-\lambda A)=0$, orIts discriminant $(\mathrm{tr}C{)}^2-4\det C$ is positive, negative, or zero in the three cases above. For the well-posed first-order evolution meaning of hyperbolicity, real eigenvalues must also admit a complete eigenbasis: a repeated but diagonalizable pair is non-strictly hyperbolic rather than a defective parabolic system. Distinct real roots suffice for strict hyperbolicity.

For the given shallow water equations, subtracting $u$ times mass conservation from momentum conservation and dividing by $h>0$ gives the primitive-variable systemThe characteristic polynomial is $(u-\lambda {)}^2-h$, soThey are real and distinct whenever $h>0$, proving strict hyperbolicity.