Discussion
Let \(Edu^{2} + 2Fdu\,dv + Gdv^{2}\) and \(Ldu^{2} + 2Mdu\,dv + Ndv^{2}\) be the first and second fundamental forms respectively of a surface patch in \(\mathbf{R}^{3}\). If \(X = \begin{bmatrix} E & F \\ F & G \end{bmatrix}\) and \(Y = \begin{bmatrix} L & M \\ M & N \end{bmatrix}\), then :
Answer: C
\(X^{-1}Y\) may not be symmetric, but has real eigenvalues.
\(X^{-1}Y\) is the matrix of the shape (Weingarten) operator in the basis \(\{\sigma_{u}, \sigma_{v}\}\). That basis need not be orthonormal, so the matrix need not be symmetric. But the shape operator is self-adjoint with respect to the first fundamental form, so its eigenvalues, the principal curvatures, are real.
It need not be non-singular: its determinant is the Gaussian curvature \(K = \det Y / \det X\), which is 0 on a cylinder or plane. So A and D fail, and B fails because the matrix is generally not symmetric.
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