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  4. Let x, y, z>1 and A=[1 log x y log x z log y x 2 log y z log z x log z y 3]. Then mid operatornameadj( operatornameadj A2. )| is equal to

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Q. Let $x, y, z>1$ and $A=\begin{bmatrix}1 & \log _x y & \log _x z \\ \log _y x & 2 & \log _y z \\ \log _z x & \log _z y & 3\end{bmatrix}$. Then $\mid \operatorname{adj}\left(\operatorname{adj} A^2\right. )|$ is equal to

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Solution:

$|A|=\frac{1}{\log x \cdot \log y \cdot \log z} \begin{vmatrix} \log x & \log y & \log z \\ \log x & 2 \log y & \log z \\ \log x & \log y & 3 \log z\end{vmatrix}=2$
$\Rightarrow\left|\operatorname{adj}\left(\operatorname{adj} A^2\right)\right|=\left|A^2\right|^4=2^8$