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:
Show Hint
When solving problems with matrix powers and sets, first compute the necessary powers of matrices, then use the given conditions to find the valid elements in the set.
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.
