Mathematics - Application of Derivative Question with Solution | TestHub

$D=\mathrm{diag}\left(d_1,d_2,\dots ,d_n\right)$ $\mathrm{D}=\left|\begin{array}{cccccc}{\mathrm{d}}_1& 0& 0& -& -& 0\\ 0& {\mathrm{d}}_2& 0& -& -& 0\\ 0& 0& {\mathrm{d}}_3& -& -& 0\\ -& -& -& -& -& -\\ -& -& -& -& -& -\\ 0& 0& 0& 0& -& 0\end{array}\right|$ $|\mathrm{D}|=\left|\begin{array}{cccccc}{\mathrm{d}}_1& 0& 0& -& -& 0\\ 0& {\mathrm{d}}_2& 0& -& -& 0\\ 0& 0& {\mathrm{d}}_3& -& -& 0\\ -& -& -& -& -& -\\ -& -& -& -& -& -\\ 0& 0& 0& 0& -& 0\end{array}\right|$ $={\mathrm{d}}_1{\mathrm{d}}_2{\mathrm{d}}_3\dots \dots {\mathrm{d}}_{\mathrm{n}}$
$\mathrm{adj}(\mathrm{D})=\frac{1}{{\mathrm{d}}_1{\mathrm{d}}_2{\mathrm{d}}_3\dots {\mathrm{d}}_{\mathrm{n}}}$ $\left[\begin{array}{cccccc}{\mathrm{d}}_2{\mathrm{d}}_3{\mathrm{d}}_4\cdot \mathrm{dn}& 0& 0& -& -& 0\\ 0& {\mathrm{d}}_1{\mathrm{d}}_3{\mathrm{d}}_4\dots \mathrm{dn}& 0& -& -& 0\\ 0& 0& {\mathrm{d}}_1{\mathrm{d}}_2{\mathrm{d}}_4\dots \mathrm{dn}& -& -& 0\\ -& -& -& -& -& -\\ -& -& -& -& -& -\\ 0& 0& 0& 0& -{\mathrm{d}}_1{\mathrm{d}}_2{\mathrm{d}}_3\dots {\mathrm{d}}_{\mathrm{n}-1}& \end{array}\right]$ $=\left[\begin{array}{cccccc}\frac{1}{d_1}& 0& 0& -& -& 0\\ 0& \frac{1}{d_2}& 0& -& -& 0\\ 0& 0& \frac{1}{d_3}& -& -& 0\\ -& -& -& -& -& -\\ -& -& -& -& -& -\\ 0& 0& 0& 0& -& \frac{1}{d_n}\end{array}\right]$ = $\begin{array}{rl}& \left[\begin{array}{cccccc}{\mathrm{d}}_1^{-1}& 0& 0& -& -& 0\\ 0& {\mathrm{d}}_2^{-1}& 0& -& -& 0\\ 0& 0& {\mathrm{d}}_3^{-1}& -& -& 0\\ -& -& -& -& -& -\\ -& -& -& -& -& -\\ 0& 0& 0& 0& -& {\mathrm{d}}_{\mathrm{n}}^{-1}\end{array}\right]\\ =& \mathrm{diag}\left({\mathrm{d}}_1^{-1},{\mathrm{d}}_2^{-1},\dots ,{\mathrm{d}}_{\mathrm{n}}^{-1}\right)\end{array}$