Matrices & Determinants
Adjoint Powers — Finding 3n+α
nta_pyq_2023_apr
Grade None
Question:
Let for $A=\begin{pmatrix}1&2&3\\\alpha&3&1\\1&1&2\end{pmatrix}$, $|A|=2$. If $|2\,\text{adj}(2\,\text{adj}(2A))|=32^n$, then $3n+\alpha$ is equal to
Step-by-Step Solution
Key Concept: From $|A|=2$ find $\alpha=-4$. Use $|\text{adj}(\text{adj}(A))|=|A|^{(n-1)^2}$ and $|kA|=k^n|A|$.
$\alpha=-4,\ n=5$. $3n+\alpha=11$.
Correct Answer: 2