Question

If \[ A=\begin{bmatrix}2 & 3\\3 & 5\end{bmatrix}, \] then find the value of \[ \left|A^{2025}-3A^{2024}+A^{2023}\right|. \]

Hint:

For determinant-based matrix problems:
Reduce high powers using Cayley–Hamilton theorem
Factor expressions before applying determinant properties
If \(|A|=1\), then \(|A^n|=1\) for all \(n\)

Answer 1

Correct Answer: 16

Concept: To handle large powers of a matrix, we apply the Cayley–Hamilton theorem. Also recall the following determinant properties:
\(|AB| = |A||B|\)
\(|kA| = k^n |A|\) for an \(n \times n\) matrix
\(|A^n| = |A|^n\)
Step 1: Determine the characteristic equation of matrix \(A\). \[ \text{Trace}(A) = 2 + 5 = 7, \quad |A| = 2 \cdot 5 - 3 \cdot 3 = 1 \] Thus, the characteristic equation is: \[ \lambda^2 - 7\lambda + 1 = 0 \] By the Cayley–Hamilton theorem: \[ A^2 - 7A + I = 0 \] \[ \Rightarrow A^2 = 7A - I \quad \cdots (1) \]
Step 2: Factor the given matrix expression. \[ A^{2025} - 3A^{2024} + A^{2023} = A^{2023}(A^2 - 3A + I) \]
Step 3: Simplify \(A^2 - 3A + I\) using equation (1). \[ A^2 - 3A + I = (7A - I) - 3A + I = 4A \] Hence, \[ A^{2025} - 3A^{2024} + A^{2023} = 4A^{2024} \]
Step 4: Evaluate the determinant. \[ \left|4A^{2024}\right| = 4^2 \left|A^{2024}\right| = 16\,|A|^{2024} \] Since \(|A| = 1\), \[ \left|4A^{2024}\right| = 16 \]