If x, y & z are non zero real numbers, the inverse of matrix $A=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$ is :

$\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$

$A=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]$

finding cofactors

$C_{11} =(-1)^{1+1} y z =y z $

$C_{12} =(-1)^{1+2} × 0 = 0$

$C_{13} =(-1)^{1+3} × 0 = 0$

$C_{21} =(-1)^{21} × 0 =0$

$C_{22} =(-1)^{2+2} x z = x z$

$C_{23} =(-1)^{2+3} × 0 =0$

$C_{31} =(-1)^{3+1} × 0 =0$

$C_{32} =(-1)^{3+2} × 0 =0$

$C_{33} =(-1)^{3+3} × xy =xy$

so adj A = $\left[\begin{array}{lll}C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33}\end{array}\right]^{T}$

$=\left[\begin{array}{lll}C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33}\end{array}\right]$

Adj A = $\left[\begin{array}{ccc}y z & 0 & 0 \\ 0 & x z & 0 \\ 0 & 0 & x y\end{array}\right]$

|A| = xyz

so  $A^{-1} = \frac{1}{|A|} A d j A=\frac{1}{x y z}\left[\begin{array}{ccc}y z & 0 & 0 \\ 0 & x z & 0 \\ 0 & 0 & x y\end{array}\right]$

$A^{-1}=\left[\begin{array}{ccc}x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & z^{-1}\end{array}\right]$