Question:
If $A=\left[\begin{array}{cc}\ln x & -1 \\ -\ln x & 2\end{array}\right]$ and if $\operatorname{det}(A)=2$, then $x=$ _______
Solution:
Given:
$A=\left[\begin{array}{cc}\ln x & -1 \\ -\ln x & 2\end{array}\right]$
$\operatorname{det}(A)=2$
Now,
$|A|=2$
$\Rightarrow\left|\begin{array}{cc}\ln x & -1 \\ -\ln x & 2\end{array}\right|=2$
$\Rightarrow 2 \ln x-\ln x=2$
$\Rightarrow \ln x=2$
$\Rightarrow x=\mathrm{e}^{2}$
Hence, $x=\underline{e}^{2}$.
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