JEE Main 2026 (Online) 22nd January Morning ShiftQuestion 1
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 $\det(\operatorname{adj}(2\operatorname{adj}(A+I)))=(2)^{\alpha}\cdot(3)^{\beta}\cdot(11)^{\gamma}$, $\alpha,\beta,\gamma$ are non-negative integers, then $\alpha+\beta+\gamma$ is equal to $\_\_\_\_\_\_$ .
Matrices
Answer$18$
Solution
Skew-symmetric $A=\begin{bmatrix}0&-3&3\\3&0&5\\-3&-5&0\end{bmatrix}$ from the two given products. $\det(I+A)=1+9+9+25=44$. $\det(\operatorname{adj}(2\operatorname{adj}B))=64(\det B)^4=2^6\cdot44^4=2^{14}\cdot11^4$, so $\alpha+\beta+\gamma=14+0+4=18$.