Q78 / 150 · MHT CET 2022 (08 Aug Shift 1) · Mathematics

If $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 3\end{array}\right]$ and $B=\operatorname{adj} A, C=54$, then $\frac{|\operatorname{adj} B|}{|C|}=$

  • A
    5
  • B
    25
  • C
    -1
  • Correct answer
    D
    1

Solution

$\begin{aligned} & A=\left|\begin{array}{ccc}1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0\end{array}\right| \\ & \Rightarrow|A|=1 \times(0+3)+1 \times(0+6)+1 \times(0-4)=5 \\ & \because B=\operatorname{adj} A \\ & \Rightarrow|B|=|\operatorname{adj} A|=|A|^2=25 \\ & \Rightarrow|\operatorname{adj} B|=|B|^2=625 \\ & \because C=5 A \\ & \Rightarrow|C|=|5 A|=5^3|A|=125 \times 5=625 \\ & \text { now } \frac{|\operatorname{adj} B|}{|C|}=\frac{625}{625}=1\end{aligned}$

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

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