Let x, y, z > 1 and A = [ cases 1 & _ x y & _ x z _ y x & 2 & _ y z...

Given:

A = \left[\begin{cases}1 & \log_{x}y & \log_{x}z \\ \log_{y}x & 2 & \log_{y}z \\ \log_{z}x & \log_{z}y & 3\end{cases}\right]

⇒ A = \left[\begin{cases}1 & \dfrac{\log y}{\log x} & \dfrac{\log z}{\log x} \\ \dfrac{\log x}{\log y} & 2 & \dfrac{\log z}{\log y} \\ \dfrac{\log x}{\log z} & \dfrac{\log y}{\log z} & 3\end{cases}\right]

⇒ A = \dfrac{1}{logxlogylogz}\left[\begin{cases}\log x & \log y & \log z \\ \log x & 2\log y & \log z \\ \log x & \log y & 3\log z\end{cases}\right]

⇒ \left|A\right| = \dfrac{1}{\log x \cdot \log y \cdot \log z}\left|\begin{cases}\log x & \log y & \log z \\ \log x & 2\log y & \log z \\ \log x & \log y & 3\log z\end{cases}\right|

⇒ \left|A\right| = \dfrac{\log x \cdot \log y \cdot \log z}{\log x \cdot \log y \cdot \log z}\left|\begin{cases}1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 3\end{cases}\right|

⇒ \left|A\right| = \left|\begin{cases}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 2\end{cases}\right|

$⇒ \left|A\right| = 2$

Now,

\left|adj \left(adj {A}^{2}\right)\right| = {\left|adj {A}^{2}\right|}^{2} = {\left|{A}^{2}\right|}^{4} = {\left|A\right|}^{8} = {2}^{8}$