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If $A$ is a $3 \times 3$ matrix such that $|5 \cdot \operatorname{adj} A|=5$, then $|A|$ is equal to
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Verified Answer
The correct answer is:
$\pm 1 / 5$
Given, $A$ is a $3 \times 3$ matrix and $|5 \cdot \operatorname{Adj}(A)|=5$
$$
\begin{aligned}
& |5 \cdot \operatorname{adj}(A)|=5 \Rightarrow 5^3|\operatorname{adj}(A)|=5 \\
\Rightarrow & |\operatorname{adj}(A)|=\frac{1}{5^2} \Rightarrow|A|^{3-1}=\frac{1}{5^2} \\
\Rightarrow & \quad|A|^2=\left(\frac{1}{5}\right)^2 \Rightarrow|A|= \pm \frac{1}{5}
\end{aligned}
$$
$$
\begin{aligned}
& |5 \cdot \operatorname{adj}(A)|=5 \Rightarrow 5^3|\operatorname{adj}(A)|=5 \\
\Rightarrow & |\operatorname{adj}(A)|=\frac{1}{5^2} \Rightarrow|A|^{3-1}=\frac{1}{5^2} \\
\Rightarrow & \quad|A|^2=\left(\frac{1}{5}\right)^2 \Rightarrow|A|= \pm \frac{1}{5}
\end{aligned}
$$
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