Q80 / 180 · KCET 2010 · Mathematics

If $A$ is a $3 \times 3$ non-singular matrix and if $|\mathrm{A}|=3$, then $\left|(2 \mathrm{~A})^{-1}\right|$ is

  • A
    24
  • B
    3
  • C
    $\frac{1}{3}$
  • Correct answer
    D
    $\frac{1}{24}$

Solution

Given, $|A|_{3 \times 3} \neq 0$ and $|A|=3$ Then, $$ \begin{aligned} &\left|(2 \mathrm{~A})^{-1}\right|=\left|\frac{1}{2 \mathrm{~A}}\right|=\frac{1}{|2 \mathrm{~A}|} \\ &=\frac{1}{(2)^{3}} \cdot \frac{1}{|\mathrm{~A}|} \quad\left(\because|\mathrm{aA}|=\mathrm{a}^{3}|\mathrm{~A}|\right) \\ &=\frac{1}{8} \cdot \frac{1}{3}=\frac{1}{24} \end{aligned} $$

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

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