Q69 / 150 · MHT CET 2023 (10 May Shift 1) · Mathematics

If $B=\left[\begin{array}{lll}1 & \alpha & 2 \\ 1 & 2 & 2 \\ 2 & 3 & 3\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $\mathrm{A}$ and $|\mathrm{A}|=5$, then $\alpha$ is equal to

  • A
    25
  • Correct answer
    B
    27
  • C
    $3 \sqrt{3}$
  • D
    5

Solution

Using $|\operatorname{Adj} A|=|A|^{n-1}$ But $\mathrm{B}=\operatorname{Adj}(\mathrm{A})$...['Given] $\therefore \quad|\mathrm{B}|=|\mathrm{A}|^2$ $\begin{aligned} & \Rightarrow\left|\begin{array}{lll}1 & \alpha & 2 \\ 1 & 2 & 2 \\ 2 & 3 & 3\end{array}\right|=|\mathrm{A}|^2 \\ & \Rightarrow \alpha-2=5^2 \\ & \Rightarrow \alpha-2=25 \\ & \Rightarrow \alpha=27\end{aligned}$

Hence, the correct answer is option: 'B'.

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