Let S = n : 1 ⩽ n ⩽ 50 and n is odd . Let a ∈ S and A = [ cases 1 &...

We know that

$\left|AdjA\right| = {\left|A\right|}^{n - 1}$

Here $n$ is $3 \times 3$ matrix

so \left|AdjA\right| = {\left|A\right|}^{3 - 1} = {\left|A\right|}^{2}</p><p>Now find \left|A\right| = \left|\begin{cases}1 & 0 & a \\ - 1 & 1 & 0 \\ - a & 0 & 1\end{cases}\right|

$= 1 + {a}^{2}$

Now \left|AdjA\right| = {\left|A\right|}^{2} = {\left(1 + {a}^{2}\right)}^{2}$</p><p>Now </p><p>$\mathop{Σ}_{a \in S} det \left(adj A\right) = Σ{\left(1 + {a}^{2}\right)}^{2} = 100\lambda$</p><p>$⇒ {2}^{2} + {4}^{2} . . . + {48}^{2} + {50}^{2} = 100\lambda$</p><p>$⇒ 4\left({1}^{2} + {2}^{2} . . . + {25}^{2}\right) = 100\lambda$</p><p>$⇒ 4 \times \dfrac{25 \times 26 \times 51}{6} = 100\lambda$</p><p>$⇒ 221 = \lambda