Question

Let $A$ be a $3\times 3$ square matrix and matrices $B,C$ and $D$ are related such that $B=adj\left(A\right),$ $C=adj\left(adjA\right)$ and $D=\left(adj\right(adj\left(adjA\right)\left)\right).$ If $det\left(adj\right(adj\left(adj\right(adjABCD\left)\right)\left)\right))=|A{|}^k,$ then the value of $k$ is

Answer 1

$B=adj\left(A\right),C=adj\left(adj\right(A\left)\right),D=adj\left(adj\right(adj\left(A\right)\left)\right)$
Let $ABCD=E$
Then, $|adj\left(adj\right(adj\left(adj\right(E\left)\right))|=|E{|}^{{(3-1)}^4}=|E{|}^{16}$

Now, $|E{|}^{16}=|ABCD{|}^{16}$
$=|A{|}^{16}|B{|}^{16}|C{|}^{16}|D{|}^{16}$
$=|A{|}^{16}\left(|A{|}^2{)}^{16}\right(|A{|}^4{)}^{16}(|A{|}^8{)}^{16}$
$(\because |B|=|adjA|=|A{|}^2|C|=|adjadjA|=|A{|}^{2^2}=|A{|}^4|D|=|adjadjadjA|=|A{|}^{2^3}=|A{|}^8)$
$\Rightarrow |E{|}^{16}=|A{|}^{16+32+64+128}$
$\Rightarrow |E{|}^{16}=|A{|}^k=|A{|}^{240}$
$\therefore k=240$