This problem beautifully combines concepts of eigenvalues, eigenvectors, and solving systems of linear equations. The key is to first understand the structure of the given information about matrix $A$ and then use it to determine the properties of the system $(A-3I)\mathbf{x} = \mathbf{b}$.

1. Key Concepts

• Eigenvalues and Eigenvectors: A non-zero vector $\mathbf{v}$ is an eigenvector of a square matrix $A$ if $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$. The scalar $\lambda$ is called the eigenvalue corresponding to $\mathbf{v}$.
• Linear Independence: A set of vectors is linearly independent if no vector in the set can be written as