Question

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:

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.

Answer 1

Step 1: Compute successive powers of matrix \( A \) using matrix multiplication, including \( A^2 \), \( A^3 \), and any higher powers required.
Step 2: Determine the set \( S \) by substituting the calculated powers of \( A \) into the equation \( A m^2 + A^n = 31 - A^6 \), and subsequently identify the values of \( m \).
Step 3: Calculate \( n(S) \), representing the cardinality of the set \( S \). The result is the value of \( n(S) \).