QJEEM24Ma7b2b: JEE Main 2024 Mathematics - Class 12 JEEM24Ma7b2b Solution: Let , where are column matrics, and If and is the sum of all…

$\mathrm{A}=\left[\begin{array}{lll}2 & 0 & 1 \\1 & 1 & 0 \\1 & 0 & 1\end{array}\right] \quad \mathrm{B}=\left[\mathrm{B}_1, \mathrm{~B}_2, \mathrm{~B}_3\right]$

$\mathrm{B}_1=\left[\begin{array}{l}\mathrm{x}_1 \\\mathrm{y}_1 \\\mathrm{z}_1\end{array}\right], \quad \mathrm{B}_2=\left[\begin{array}{l}\mathrm{x}_2 \\\mathrm{y}_2 \\\mathrm{z}_2\end{array}\right], \quad \mathrm{B}_3=\left[\begin{array}{l}\mathrm{x}_3 \\\mathrm{y}_3 \\\mathrm{z}_3\end{array}\right]$

$\mathrm{AB}_1=\left[\begin{array}{lll}2 & 0 & 1 \\1 & 1 & 0 \\1 & 0 & 1\end{array}\right]\left[\begin{array}{l}\mathrm{x}_1 \\\mathrm{y}_1 \\\mathrm{z}_1\end{array}\right]=\left[\begin{array}{l}1 \\0 \\0\end{array}\right]$

$\begin{gathered}\mathrm{x}_1=1, \mathrm{y}_1=-1, \mathrm{z}_1=-1 \\\mathrm{AB}_2=\left[\begin{array}{lll}2 & 0 & 1 \\1 & 1 & 0 \\1 & 0 & 1\end{array}\right]\left[\begin{array}{l}\mathrm{x}_2 \\\mathrm{y}_2 \\\mathrm{z}_2\end{array}\right]=\left[\begin{array}{l}2 \\3 \\0\end{array}\right] \\\mathrm{x}_2=2, \mathrm{y}_2=1, \mathrm{z}_2=-2 \\\mathrm{AB}_3=\left[\begin{array}{lll}2 & 0 & 1 \\1 & 1 & 0 \\1 & 0 & 1\end{array}\right]\left[\begin{array}{l}\mathrm{x}_3 \\\mathrm{y}_3 \\\mathrm{z}_3\end{array}\right]=\left[\begin{array}{l}3 \\2 \\1\end{array}\right] \\\mathrm{x}_3=2, \mathrm{y}_3=0, \mathrm{z}_3=-1 \\\mathrm{~B}=\left[\begin{array}{ccc}1 & 2 & 2 \\-1 & 1 & 0 \\-1 & -2 & -1\end{array}\right] \\\alpha=|\mathrm{B}|=3 \\\beta=1 \\\alpha^3+\beta^3=27+1=28\end{gathered}$