Let
\[
S = \left\{ m \in \mathbb{Z} : A m^2 + A^n = 31 - A^6 \right\}, \quad \text{where} \quad A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}
\]
Then \( n(S) \) is equal to:
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Solution and Explanation
Step 1: First, calculate the powers of the matrix \( A \). Use matrix multiplication to compute \( A^2 \), \( A^3 \), and higher powers as necessary.
Step 2: Solve for the set \( S \) by substituting the appropriate powers of \( A \) into the given condition \( A m^2 + A^n = 31 - A^6 \), and identify the values of \( m \).
Step 3: Finally, compute \( n(S) \), which is the number of elements in the set \( S \). Thus, the final value of \( n(S) \) is found.
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