Question

$A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix} \Rightarrow A^{2} = \begin{bmatrix} 13 & 21 \\ 21 & 34 \end{bmatrix}$

$$\begin{matrix} & \left| A^{2025} - 3A^{2024} + A^{2023} \right| \\ & \ = \left| A^{2023}\left( A^{2} - 3A + I \right) \right| \\ & \ = |A|^{2023}\left| A^{2} - 3A + I \right| \\ & \ = 1 \cdot \left| \begin{matrix} 8 & 12 \\ 12 & 20 \end{matrix} \right| = 160 - 144 = 16 \end{matrix}$$