Given : $\mathrm{A}=\left[\begin{array}{ccc}2& 1& -1\\ 0& 1& 4\\ 0& 0& 3\end{array}\right]$

we have, $\mathrm{A}=\left[\begin{array}{ccc}2& 1& -1\\ 0& 1& 4\\ 0& 0& 3\end{array}\right]$

As we know, that

adj(adjA) = |A|^n–2.A

where n is

order of matrix

∴ Here, adj(adjA) = |A|^3–2 . A

[∵  order of matrix A = 3 × 3]

⇒ adj(adjA) = |A| . A

∴ |A| = 2(3 – 0) – 1(0) – 1(0)

|A| = 6

∴ adj(adjA) = 6 . A

$=6\left[\begin{array}{ccc}2& 1& -1\\ 0& 1& 4\\ 0& 0& 3\end{array}\right]$

trace of matrix : the sum of elements on the main diagonal of matrix.

$\therefore \mathrm{tr}(\mathrm{adj}(\mathrm{adj}\mathrm{A}\left)\right)=\mathrm{tr}\left(6\left[\begin{array}{ccc}2& 1& -1\\ 0& 1& 4\\ 0& 0& 3\end{array}\right]\right)$

$6+\mathrm{r}\left[\begin{array}{ccc}2& 1& -1\\ 0& 1& 4\\ 0& 0& 3\end{array}\right]$

= 6(2 + 1 + 3)

= 6(6)

⇒ tr(adj(adjA)) = 36

∴ option C is correct

⇒ tr(adj(adjA)) = 36