Matrices & Determinants
Cayley-Hamilton — det of Matrix Polynomial
nta_pyq_2026_jan
Grade 12
Question:
If $A=\begin{bmatrix}2&3\\3&5\end{bmatrix}$, then the determinant of the matrix $\left(A^{2025}-3A^{2024}+A^{2023}\right)$ is
Step-by-Step Solution
Key Concept: $A^{2025}-3A^{2024}+A^{2023}=A^{2023}(A^2-3A+I)$. $\det(A^{2023})=\det(A)^{2023}=(10-9)^{2023}=1$. Compute $A^2-3A+I$.
$\det(A^{2025}-3A^{2024}+A^{2023})=16$.
Correct Answer: 4