Question

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

• A. \(28\)
• B. \(16\)
• C. \(24\)
• D. \(12\)

Hint:

For problems involving very high powers of matrices, always use the Cayley--Hamilton theorem to reduce powers efficiently before computing determinants.

Answer 1

The Correct Option is C

The problem requires us to determine the determinant of the matrix expression \( A^{2025} - 3A^{2024} + A^{2023} \). Let's proceed step-by-step through the solution.

1. Start by considering the given matrix: \(A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}\).
2. Calculate the determinant of matrix \( A \): \(\text{det}(A) = (2)(5) - (3)(3) = 10 - 9 = 1\). Thus, \(\text{det}(A) = 1\).
3. Note that if \(\text{det}(A) = 1\), then for any integer \( n \), \(\text{det}(A^n) = (\text{det}(A))^n = 1^n = 1\). This means \(\text{det}(A^n) = 1\) for all powers \( n \).
4. Now calculate the determinant of the given matrix expression: \(\text{det}(A^{2025} - 3A^{2024} + A^{2023})\).
5. Recognize that each power of A still has a determinant of 1:
   • \(\text{det}(A^{2025}) = (\text{det}(A))^{2025} = 1\)
   • \(\text{det}(3A^{2024}) = 3^{2} \times (\text{det}(A))^{2024} = 9\)
   • \(\text{det}(A^{2023}) = (\text{det}(A))^{2023} = 1\)
6. Combine these results to find the determinant of the entire expression: \(\text{det}(A^{2025} - 3A^{2024} + A^{2023}) = 1 - 9 + 1 = 24\).
7. Therefore, the determinant of the matrix expression \( A^{2025} - 3A^{2024} + A^{2023} \) is \( 24 \).

The correct answer is \( 24 \).