Question

If $A=\left[\begin{array}{cc}\ln x& -1\\ -\ln x& 2\end{array}\right]$ and if det (A) = 2, then x = ___________.

Answer 1

Given:
$A=\left[\begin{array}{cc}\ln x& -1\\ -\ln x& 2\end{array}\right]$
det (A) = 2



$$\begin{gathered} \mathrm{Now}, \\ \left|A\right|=2 \\ \Rightarrow \left|\begin{array}{cc}\ln x& -1\\ -\ln x& 2\end{array}\right|=2 \\ \Rightarrow 2\ln x-\ln x=2 \\ \Rightarrow \ln x=2 \\ \Rightarrow x={\mathrm{e}}^2 \end{gathered}$$

Hence, x = e².