This problem elegantly combines the concept of the Characteristic Equation of a matrix with the powerful Cayley-Hamilton Theorem.

1. Key Concept: Cayley-Hamilton Theorem

The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation. For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, its characteristic equation is given by $\det(A - \lambda I) = 0$, which expands to: $\lambda^2 - (\text{Tr}(A))\lambda + \det(A) = 0$ where $\text{Tr}(A)$ is the trace of matrix $A$ (sum of diagonal elements, $a+d$)