If $$A=\begin{bmatrix}2 & 3 \\3 & 5 \end{bmatrix}$$, then the determinant of the matrix $$ (A^{2025}-3A^{2024}+ A^{2023})$$ is

Characteristic Equation of A:

$$\lambda^2 - \text{Tr}(A)\lambda + |A| = 0$$

$$\text{Tr}(A) = 2 + 5 = 7$$

$$|A| = (2)(5) - (3)(3) = 10 - 9 = 1$$

$$\implies A^2 - 7A + I = 0 \implies A^2 + I = 7A$$

$$X = A^{2025} - 3A^{2024} + A^{2023} = A^{2023}(A^2 - 3A + I)$$

Substitute $$A^2 + I = 7A$$:

$$X = A^{2023}(7A - 3A) = A^{2023}(4A) = 4A^{2024}$$

$$|X| = |4A^{2024}| = 4^2 \cdot |A|^{2024}$$

Since $$A$$ is a $$2 \times 2$$ matrix, $$|4M| = 4^2|M|$$.

$$|X| = 16 \cdot (1)^{2024} = \mathbf{16}$$