What is the process for solving a system of first-order ordinary differential equations using matrix methods as described in the chapter?

To solve a system of first-order ordinary differential equations using matrix methods, first find the eigenvalues and associated eigenvectors of the matrix representing the system. Then, express the solution in terms of these eigenvalues and eigenvectors, applying boundary conditions to determine any constants involved in the solution.

The process involves the following steps: 1. Write the system of equations in matrix form as F(x) = AF(x). 2. Identify the eigenvalues and eigenvectors of the matrix A. 3. Formulate the solution as a linear combination of the eigenvectors multiplied by exponentials of their corresponding eigenvalues. 4. Use the given boundary conditions to solve for any constants in the solution.