Let $A=\begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$, $B=\begin{bmatrix} B_{1} & B_{2}…

Let $A=\begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$, $B=\begin{bmatrix} B_{1} & B_{2} & B_{3} \end{bmatrix}$, where $B_{1}$, $B_{2}$, $B_{3}$ are column matrices, and $AB_{1}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$, $AB_{2}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$, $AB_{3}=\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$. If $\alpha=|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^{3}+\beta^{3}$ is equal to

Solution

Given: $A=\begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$, $B=\begin{bmatrix} B_{1} & B_{2} & B_{3} \end{bmatrix}$, $B_{1}=\begin{bmatrix} x_{1} \\ y_{1} \\ z_{1} \end{bmatrix}$, $B_{2}=\begin{bmatrix} x_{2} \\ y_{2} \\ z_{2} \end{bmatrix}$ and $B_{3}=\begin{bmatrix} x_{3} \\ y_{3} \\ z_{3} \end{bmatrix}$. $\Rightarrow AB_{1}=\begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}\begin{bmatrix} x_{1} \\ y_{1} \\ z_{1} \end{bmatrix}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ $\Rightarrow \begin{bmatrix} 2x_{1}+0+z_{1} \\ x_{1}+y_{1}+0 \\ x_{1}+0+z_{1} \end{bmatrix}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ $\Rightarrow 2x_{1}+z_{1}=1$, $x_{1}+y_{1}=0$, $x_{1}+z_{1}=0$ $\Rightarrow x_{1}=1$, $y_{1}=-1$, $z_{1}=-1$ $\Rightarrow AB_{2}=\begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}\begin{bmatrix} x_{2} \\ y_{2} \\ z_{2} \end{bmatrix}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$ $\Rightarrow AB_{2}=\begin{bmatrix} 2x_{2}+z_{2} \\ x_{2}+y_{2} \\ x_{2}+z_{2} \end{bmatrix}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$ $\Rightarrow 2x_{2}+z_{2}=2, x_{2}+y_{2}=3, x_{2}+z_{2}=0$ $\Rightarrow x_{2}=2, y_{2}=1, z_{2}=-2$ $\Rightarrow AB_{3}=\begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_{3} \\ y_{3} \\ z_{3} \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$ $\Rightarrow AB_{2}=\begin{bmatrix} 2x_{3}+z_{3} \\ x_{3}+y_{3} \\ x_{3}+z_{3} \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}$ $\Rightarrow 2x_{3}+z_{3}=3, x_{3}+y_{3}=2, x_{3}+z_{3}=1$ $\Rightarrow x_{3}=2, y_{3}=0, z_{3}=-1$ $\Rightarrow B=\begin{bmatrix} 1 & 2 & 2 \\ -1 & 1 & 0 \\ -1 & -2 & -1 \end{bmatrix}$ $\Rightarrow \alpha=|B|=3$ $\Rightarrow \beta=1$ $\Rightarrow \alpha^{3}+\beta^{3}=27+1=28$

Asked in: JEE Main 2024 (27 Jan Shift 1)

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