how to find determinant of the following matrix

Question : Find determinant of the following matrix \[A=\left[\begin{matrix}1&\log_x{y}&\log_x{z}\\\log_y{x}&1&\log_y{z}\\\log_z{x}&\log_z{y}&1\\\end{matrix}\right]\] Solution :Given matrix is\[A=\left[\begin{matrix}1&\log_x{y}&\log_x{z}\\\log_y{x}&1&\log_y{z}\\\log_z{x}&\log_z{y}&1\\\end{matrix}\right]\] Using formula of determinant of 3 by 3 matrix, It’s determinant is given as \[determinant(A)=\left|\begin{matrix}1&\log_x{y}&\log_x{z}\\\log_y{x}&1&\log_y{z}\\\log_z{x}&\log_z{y}&1\\\end{matrix}\right|\] \[determinant(A)=1\left(1-\log_z{y}\log_y{z}\right)-\log_x{y}\left(\log_y{x}-\log_z{x}\log_y{z}\right)+\log_x{z}\left(\log_y{x}\log_z{y}-\log_z{x}\right)\] \[determinant(A)=1-\log_z{y}\log_y{z}-\log_x{y}\log_y{x}+\log_x{y}\log_z{x}\log_y{z}+\log_x{z}\log_y{x}\log_z{y}-\log_x{z}\log_z{x}\] By the property \[\log_a{b}=\frac{\log_e{b}}{\log_e{a}}\]we shall obtain \[\log_z{y}\log_y{z}=1,\log_x{y}\log_y{x}=1, \log_x{y}\log_z{x}\log_y{z}= 1, \log_x{z}\log_y{x}\log_z{y}…

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