If $D=diag\left(d_1,d_2,d_3,\mathrm{.....},d_n\right)$ where $d_i\ne 0$ for all $i=1,2,\mathrm{......},n$ ,be a diagonal matrix ,then $D^{-1}$ is equal to

$letD=diag\left(d_1,d_2,d_3,\mathrm{.....},d_n\right)then$

$\left|D\right|=d_1{d}_2\mathrm{.....}{d}_n$

$cofactorof{d}_1=d_2{d}_3\mathrm{.....}{d}_n$

$cofactorof{d}_1=d_2{d}_3\mathrm{.....}{d}_n$

$cofactor{d}_n=d_1{d}_2\mathrm{.....}{d}_{n-1}$

$Aslo{D}_{ij}=0ifi\ne j$

$now{D}^{-1}=\frac{adjD}{\left|D\right|}$

$=\frac{diag\left(d_2{d}_3\mathrm{.....}{d}_n,\mathrm{....},d_1{d}_2\mathrm{.....}{d}_{n-1}\right)}{d_1{d}_2\mathrm{.....}{d}_n}$

$=diag\left(d_{{}_1}^{-1}{d}_2^{-1}\mathrm{.....}{d}_n^{-1}\right)$