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If $A$ is a $3 \times 3$ non-singular matrix and if $|\mathrm{A}|=3$, then $\left|(2 \mathrm{~A})^{-1}\right|$ is
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2747 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{24}$
Given, $|A|_{3 \times 3} \neq 0$ and $|A|=3$
Then,
$$
\begin{aligned}
&\left|(2 \mathrm{~A})^{-1}\right|=\left|\frac{1}{2 \mathrm{~A}}\right|=\frac{1}{|2 \mathrm{~A}|} \\
&=\frac{1}{(2)^{3}} \cdot \frac{1}{|\mathrm{~A}|} \quad\left(\because|\mathrm{aA}|=\mathrm{a}^{3}|\mathrm{~A}|\right) \\
&=\frac{1}{8} \cdot \frac{1}{3}=\frac{1}{24}
\end{aligned}
$$
Then,
$$
\begin{aligned}
&\left|(2 \mathrm{~A})^{-1}\right|=\left|\frac{1}{2 \mathrm{~A}}\right|=\frac{1}{|2 \mathrm{~A}|} \\
&=\frac{1}{(2)^{3}} \cdot \frac{1}{|\mathrm{~A}|} \quad\left(\because|\mathrm{aA}|=\mathrm{a}^{3}|\mathrm{~A}|\right) \\
&=\frac{1}{8} \cdot \frac{1}{3}=\frac{1}{24}
\end{aligned}
$$
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