Let A and B be two 3 × 3 matrices such that AB = I and |A| =...

Given, $AB = I$ and $\left|A\right| = \dfrac{1}{8}$

Now by using property $adj\left(PQ\right) = \left(adj Q\right)\left(adj P\right)$ we get, \left|adj \left(B \cdot adj 2A\right)\right| = \left|adj \left(adj 2A\right)\right|\left|adj B\right| . . . \left(i\right)$</p><p>Now \left|adj\left(adj2A\right)\right| = {\left|adj2A\right|}^{2}$ $= {\left|2A\right|}^{4}$

$= {\left|{2}^{3}\right|}^{4}{\left|A\right|}^{4} = {\left({2}^{3}\right)}^{4}{\left|A\right|}^{4} = {8}^{4}{\left|A\right|}^{4} . . . \left(ii\right)$

{Property used \left|adj A\right| = {\left|A\right|}^{n - 1} and $\left|kA\right| = {k}^{n}\left|A\right|$ where $n$ is order of matrix}

Now \left|adj B\right| = {\left|B\right|}^{2} . . . . . . . . . . . . . ( iii )$</p><p>Now putting the value of equation \left(ii\right) \text{ and } \left(iii\right)$ in eq \left(i\right)$</p><p>we get \left|adj \left(B \cdot adj 2A\right)\right| = {8}^{4}{\left|A\right|}^{4}{\left|B\right|}^{2}$

Now using\left|A\right| = \dfrac{1}{\left|B\right|}$as$AB = I$, we get</p><p>\left|adj \left(B \cdot adj 2A\right)\right| = {8}^{4}{\left|A\right|}^{4} \times \dfrac{1}{{\left|A\right|}^{2}}$ $= {8}^{4} \times {\left|A\right|}^{2} = {8}^{4} \times \dfrac{1}{{8}^{2}} = 64$