Let $S = \left\{m \in \mathbb{Z} : A^{m^2} + A^m = 3I - A^{-6}\right\}$, where $A = \begin{bmatrix}2 & -1 \\ 1 & 0\end{bmatrix}$. Then $n(S)$ is equal to ___
Step-by-Step Solution
Key Concept: Find the characteristic polynomial of $A$ (it equals $(\lambda-1)^2 = 0$), write $A = I + N$ where $N^2 = 0$, and use this to compute any power of $A$ efficiently.
Characteristic equation: $(\lambda-1)^2 = 0$, so $A = I+N$, $N^2 = 0$. $A^m = I + mN$. $A^{-1} = I - N$, $A^{-6} = I - 6N$. Substituting: $(m^2+m)N = 6N$. Since $N \neq 0$: $m^2+m-6=0 \Rightarrow m = 2$ or $m = -3$. $n(S) = 2$.
Correct Answer: 2