Mathematics Question from JEE Main 2026 · 22 January, Shift 1

Let $A$ be a $3 \times 3$ matrix such that $A + A^{T} = O$. If

$$A \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}, \quad A^{2} \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$$

and

$$\det\!\big(\operatorname{adj}(2\,\operatorname{adj}(A+I))\big) = 2^{\alpha}\,3^{\beta}\,11^{\gamma}$$

where $\alpha, \beta, \gamma$ are non-negative integers, then the value of $\alpha + \beta + \gamma$ is _____.