If $A$ is a $3\times3$ skew-symmetric matrix, find $\det(2A+I)$ given $\det(A+I)=2$
Step-by-Step Solution
Key Concept: Skew-symmetric: eigenvalues are $0,\pm bi$; characteristic polynomial relates $\det(A+I)$ and $\det(2A+I)$
$b=1$. Eigenvalues of $A$: $0,i,-i$. $\det(2A+I)=(1)(1+2i)(1-2i)=1+4=5$.
Correct Answer: 5