1. **State the problem:** The question requires finding the general solution to the system of differential equations $\vec{y}' = A \vec{y}$ where $A$ is a given matrix. 2. **What is required:** It asks to use the method of eigenvalues and eigenvectors to solve the system. 3. **Key steps involved:** - Find eigenvalues by solving $\det(A - \lambda I) = 0$. - Find eigenvectors corresponding to each eigenvalue. - If eigenvalues have multiplicity greater than one, find generalized eigenvectors. - Write the general solution as a linear combination of eigenvectors and generalized eigenvectors multiplied by exponential functions. 4. **Purpose:** This method transforms the system into simpler components that can be solved explicitly. 5. **Summary:** The question requires you to find eigenvalues, eigenvectors, generalized eigenvectors if needed, and then write the full general solution of the system using these components.