Matrices & Determinants
Skew-Symmetric Matrix — det(adj(2adj(A+I)))
nta_pyq_2026_jan
Grade 12
Question:
Let $A$ be a $3\times3$ matrix such that $A+A^T=O$. If $A\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}3\\3\\2\end{bmatrix}$, $A^2\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}-3\\19\\-24\end{bmatrix}$ and $\det(\text{adj}(2\,\text{adj}(A+I)))=(2)^\alpha\cdot(3)^\beta\cdot(11)^\gamma$, $\alpha,\beta,\gamma$ are non-negative integers, then $\alpha+\beta+\gamma$ is equal to _____.
Step-by-Step Solution
Key Concept: $A$ is skew-symmetric: $A=\begin{bmatrix}0&-c&b\\c&0&-a\\-b&a&0\end{bmatrix}$. From $A\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}3\\3\\2\end{bmatrix}$: $c=3$, $a+b=-2$. From $A^2\begin{bmatrix}1\\-1\\0\end{bmatrix}$: $b=3$, $a=-5$.
$\alpha+\beta+\gamma=18$.
Correct Answer: 18