Matrices & Determinants
Cayley-Hamilton — (det(adj(A³−B³)))²
nta_pyq_2026_jan
Grade 12
Question:
For some $\alpha,\beta\in\mathbb{R}$, let $A=\begin{bmatrix}\alpha&2\\1&2\end{bmatrix}$ and $B=\begin{bmatrix}1&1\\1&\beta\end{bmatrix}$ be such that $A^2-4A+2I=B^2-3B+I=O$. Then $\left(\det\!\left(\text{adj}\!\left(A^3-B^3\right)\right)\right)^2$ is equal to _____.
Step-by-Step Solution
Key Concept: From $A^2-4A+2I=O$: characteristic equation $\lambda^2-4\lambda+2=0\Rightarrow\alpha+2=4\Rightarrow\alpha=2$. From $B^2-3B+I=O$: $\beta+1=3\Rightarrow\beta=2$. $A^3=14A-8I$, $B^3=8B-3I$.
$(\det(\text{adj}(A^3-B^3)))^2=225$.
Correct Answer: 225