Matrix Adjoint Determinant (Jee Mains)

Mole Concept & Percentage Composition

Hard

Matrix Adjoint Determinant

Let A be a 3 × 3 matrix such that $\left| {\;adj\;(adj\;(adj\;A))} \right|$ = 81. If $S = \left\{ {n \in {\Bbb Z}:{{\left( {\left| {adj(adj\;A)} \right|} \right)}^{\frac{{{{(n - 1)}^2}}}{2}}} = \;|A{|^{\left( {3{n^2} - 5n - 4} \right)}}} \right\}$ , then $\sum\limits_{n\; \in \;S} {\left| {{A^{\left( {{n^2} + n} \right)}}} \right|}$ is equal to

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