Let A be a 3 × 3 matrix such that |adj(adj(adj . A))| = 12 ^ 4 ....

Given equation is

$\left|adj\left(adj\left(adj . A\right)\right)\right| = {12}^{4}$

$⇒ {\left|A\right|}^{( n - 1{)}^{3}} = {12}^{4}$

$∵ n = 3$

$⇒ {\left|A\right|}^{( 3 - 1{)}^{3}} = {12}^{4}$

$⇒ {\left|A\right|}^{8} = {12}^{4}$

$⇒ {\left|A\right|}^{2} = 12$

$⇒ \left|A\right| = 2\sqrt{3}$

Now, \left|{A}^{- 1}adjA\right| = \left|{A}^{- 1}\right| . \left|adjA\right| = \dfrac{1}{\left|A\right|} \times {\left|A\right|}^{3 - 1}$</p><p>$= \left|A\right| = 2\sqrt{3}$