Mathematics - Matrix Question with Solution | TestHub

$\begin{array}{rl}& |\mathrm{A}|=3,|\mathrm{B}|=2\\ & \left|{\mathrm{A}}^{\mathrm{T}}\mathrm{A}(\mathrm{adj}(2\mathrm{A}){)}^{-1}(\mathrm{adj}(4\mathrm{B}))(\mathrm{adj}(\mathrm{AB}){)}^{-1}{\mathrm{AA}}^{\mathrm{T}}\right|\end{array}$ $\begin{array}{c}=3\times 3\times \mid (\mathrm{adj}(2\mathrm{A}{)}^{-1}|\times |\mathrm{adj}(4\mathrm{B})|\times |(\mathrm{adj}(\mathrm{AB}){)}^{-1}\mid \times 3\times 3\\ \downarrow \downarrow \downarrow \\ \frac{1}{|\mathrm{adj}(2\mathrm{A})|}{2}^{12}\times 2^2\frac{1}{|\mathrm{adj}(\mathrm{AB})|}\end{array}$
$\begin{array}{rl}& =\frac{1}{2^6|\mathrm{adj}\mathrm{A}|}=\frac{1}{|\mathrm{adj}\mathrm{B}\cdot \mathrm{adj}\mathrm{A}|}\\ & =\frac{1}{2^6\cdot 3^2}=\frac{1}{2^2\cdot 3^2}\\ & =3^4\cdot \frac{1}{2^6\cdot 3^2}\cdot 2^{12}\cdot 2^2\cdot \frac{1}{2^2\cdot 3^2}=64\\ & \end{array}$