Mathematics Question from JEE Main 2024 · 27 January, Shift 1

Let $A$ denote the matrix $A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$, and $B = [B_1, B_2, B_3]$ where $B_1$, $B_2$, and $B_3$ are column matrices such that:

$$AB_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad AB_2 = \begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}, \quad AB_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$$

If $\alpha = |B|$ and $\beta$ is the sum of all diagonal elements of $B$, then $\alpha^3 + \beta^3$ is equal to: