Let S = $=\{\sqrt{\mathrm{n}}:1\le \mathrm{n}\le 50\text{ and }\mathrm{n}\text{ is odd }\}$Let $\mathrm{a}\in \mathrm{S}\text{ and }\mathrm{A}=\left[\begin{array}{ccc}1& 0& \mathrm{a}\\ -1& 1& 0\\ -\mathrm{a}& 0& 1\end{array}\right]$If  $\sum _{\mathrm{a}\in \mathrm{S}} \det (\mathrm{adj}\mathrm{A})=100\mathrm{\lambda },\text{ then }\mathrm{\lambda }$is equal to

$\begin{array}{l}\mathrm{S}=\{\sqrt{\mathrm{n}}:1\le \mathrm{n}\le 50\text{ and }\mathrm{n}\text{ is odd }\}\\ =\{\sqrt{1},\sqrt{3},\sqrt{5}\dots \dots \dots \sqrt{49}\},25\text{ terms }\\ \left|\mathrm{A}\right|=1+{\mathrm{a}}^2\\ \sum _{\mathrm{a}\in \mathrm{S}} \det \left(\mathrm{adjA}\right)=\sum _{\mathrm{a}\in \mathrm{S}} |\mathrm{A}{|}^2=\sum {\left(1+{\mathrm{a}}^2\right)}^2\\ ={\left(1+1\right)}^2+{\left(1+3\right)}^2+{\left(1+5\right)}^2+.....+{\left(1+49\right)}^2\\ \Rightarrow 22100=100\mathrm{\lambda }\\ \mathrm{\lambda }=221\end{array}$