Let A be a 3 3 matrix such that |adj(adj(adjA))|=81.

Question

Let $A$ be a $3	imes 3$ matrix such that $|adj(adj(adjA))|=81$.

If $S={n\in Z:(|adj(adjA)|{)}^{\frac{(n-1{)}^{2}}{2}}=|A{|}^{(3{n}^{2}-5n-4)}}$, then $\sum _{n\in S}|{A}^{({n}^{2}+n)}|$ is equal to :

Accepted Answer

$\begin{gathered} |adj(adj(adjA))|=81 \ =|A{|}^{(n-1{)}^{3}}=(3{)}^{4}\Rightarrow |A{|}^{8}={3}^{4}\Rightarrow |A|={3}^{1/2} \ |adj(adjA){|}^{\frac{(n-1{)}^{2}}{2}}=|A{|}^{(3{n}^{2}-5n-4)} \ {[|A{|}^{(n-1{)}^{2}}]}^{\frac{(n-1{)}^{2}}{2}}=|A{|}^{3{n}^{2}-5n-4} \ |A{|}^{2(n-1{)}^{2}}=|A{|}^{3{n}^{2}-5n-4} \ \Rightarrow 2(n-1{)}^{2}=3{n}^{2}-5n-4 \ \, {n}^{2}-n-6=0 \ \Rightarrow n=-2,3 \ \sum _{x\leftarrow 5}|{A}^{{n}^{2}+n}|=|{A}^{2}|+|{A}^{12}| \ =3+{3}^{6}=732 \end{gathered}$

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