Let x,y,z1 and [A=pmatrix1 & _xy & _xz _yx & 2 & _yz _zx & _zy & 3pmatrix.] Then adj(adj(…

Question

Let $x,y,z>1$ and
$$A=\begin{pmatrix}1 & \log_xy & \log_xz\ \log_yx & 2 & \log_yz\ \log_zx & \log_zy & 3\end{pmatrix}.$$
Then $\operatorname{adj}(\operatorname{adj}(A^2))$ is equal to $($a scalar multiple of $I)$, and that scalar is:

Accepted Answer

Writing all logs in terms of $\ln x,\ln y,\ln z$ and factoring, $|A|=2$ (a standard identity for this determinant pattern).

For a $3	imes3$ matrix, $\operatorname{adj}(\operatorname{adj}(A^2))=(\det A^2)^{(3-1)-1}A^2=(\det A)^2\cdot A^2$ in scalar-determinant terms; taking determinants, $|\operatorname{adj}(\operatorname{adj}(A^2))|=|A^2|^{4}=|A|^{8}=2^8$.

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