\(S = \{\sqrt n;1\le n \le50\} = a\in S\{\sqrt 1, \sqrt3, \sqrt 5,....,\sqrt{49}\}\)
\(A = \begin{bmatrix}-1 &0 &a\\-1&1&0\\-a&0&1\end{bmatrix}\)
\(adj\,A = \begin{bmatrix}1 &0 &-a\\1&a^2-1&-a\\a&0&-1\end{bmatrix}\)
\(det(adj\,A) = \begin{bmatrix}1 &0 &-a\\1&a^2-1&-a\\a&0&-1\end{bmatrix}\)
\(= (a^2 - 1)\begin{vmatrix}1&-a\\a&-1\end{vmatrix}\)
\(= (a^2 - 1) (-1 + a^2)\)
\(= (a^2 - 1)^2\)
\(\because a\in \{\sqrt 1, \sqrt3, \sqrt 5,....,\sqrt{49}\}\)
\(\therefore a^2\in \{1,3,5,....,49\}\)
\(a^2 - 1\in \{0,2,4,,....,48\}\)
\(\therefore \sum det (adj \, A) = 0^2 + 2^2 + 4^2 + ....
+48^2\)
\(= 2^2 (1^2 + 2^2 + 3^2 + .... + 24^2)\)
\(= 4\left(\frac{24(24+1)(48 +1)}{6}\right)\)
\(= 16 \times 25 \times 49\)
\(= 19600\)
\(= 196 \times 100\)
\(\therefore \lambda = 196\)