In this paper, we obtain a formula for the derivative of a determinant with respect to an eigenvalue in the modified Cholesky decomposition of a symmetric matrix, a characteristic example of a direct solution method in computational linear algebra. We apply our proposed formula to a technique used in nonlinear finite-element methods and discuss methods for determining singular points, such as bifurcation points and limit points. In our proposed method, the increment in arc length (or other relevant quantities) may be determined automatically, allowing a reduction in the number of basic parameters. The method is particularly effective for banded matrices, which allow a significant reduction in memory requirements as compared to dense matrices. We discuss the theoretical foundations of our proposed method, present algorithms and programs that implement it, and conduct numerical experiments to investigate its effectiveness.