Let $A$ be a $2 \times 2$ matrix satisfying $A^2 - 31A + I = 0$, where $I$ is the identity matrix. If $\lambda$ is an eigenvalue of $A$, the value of $\lambda^2 + \lambda^{-2}$ is ________ (answer in integer).
Answer Key:
959
Step-by-Step Solution & Explanation
Since $\lambda$ is an eigenvalue of $A$, it satisfies the characteristic equation:
$\lambda^2 - 31\lambda + 1 = 0$
Dividing through by $\lambda$ (since $\lambda \neq 0$):
$\lambda + \frac{1}{\lambda} = 31$
Squaring both sides:
$\left(\lambda + \frac{1}{\lambda}\right)^2 = 31^2 = 961$
$\lambda^2 + 2 + \frac{1}{\lambda^2} = 961 \implies \lambda^2 + \lambda^{-2} = 961 - 2 = 959$