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 | adj ( adj A 2 ) | is equal to [2023]

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Solution

Q.
$Let x, y, z > 1 and A = [\begin{matrix} 1 & \log_{x} y & \log_{x} z \\ \log_{y} x & 2 & \log_{y} z \\ \log_{z} x & \log_{z} y & 3 \end{matrix}] .$ Then $| adj (adj A^{2}) |$ is equal to [2023]

1 $4^{8}$

2 $2^{8}$

3 $2^{4}$

4 $6^{4}$

Ans.
(2)

$| A |$ $=$ $| \begin{matrix} \log_{x} x & \log_{x} y & \log_{x} z \\ \log_{y} x & 2 \log_{y} y & \log_{y} z \\ \log_{z} x & \log_{z} y & 3 \log_{z} z \end{matrix} |$

$= \frac{1}{\log x \cdot \log y \cdot \log z}$ $| \begin{matrix} \log x & \log y & \log z \\ \log x & 2 \log y & \log z \\ \log x & \log y & 3 \log z \end{matrix} |$ $[∵ \log_{a} b = \frac{\log b}{\log a}]$

$= \frac{\log x \log y \log z}{\log x \log y \log z} \times$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 3 \end{matrix} |$ $= 2$

$∴$ $| adj (adj (A^{2})) |$ $= | adj (A^{2}) |^{2}$ $= {(| A^{2} |^{2})}^{2} = | A |^{8} = 2^{8}$

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