Question

If A is an invertible matrix of order 3 and $\left|A\right|=4,\mathrm{then}\left|\mathrm{adj}\left(adjA\right)\right|$ =__________________.

Answer 1

Given:
A is an invertible matrix of order 3
$\left|A\right|$ = 4


As we know,
$$\begin{gathered} \left|\mathrm{adj}\left(\mathrm{adj}A\right)\right|={\left|A\right|}^{{\left(n-1\right)}^2},\mathrm{where}n\mathrm{is}\mathrm{the}\mathrm{order}\mathrm{of}A \\ \Rightarrow \left|\mathrm{adj}\left(\mathrm{adj}A\right)\right|={\left|A\right|}^{{\left(3-1\right)}^2}\left(\because \mathrm{Order}\mathrm{of}A\mathrm{is}3\right) \\ \Rightarrow \left|\mathrm{adj}\left(\mathrm{adj}A\right)\right|={\left|A\right|}^{{\left(2\right)}^2} \\ \Rightarrow \left|\mathrm{adj}\left(\mathrm{adj}A\right)\right|={\left|A\right|}^4 \\ \Rightarrow \left|\mathrm{adj}\left(\mathrm{adj}A\right)\right|=4^4\left(\because \left|A\right|=4\right) \\ \Rightarrow \left|\mathrm{adj}\left(\mathrm{adj}A\right)\right|=256 \end{gathered}$$


Hence, |adj (adj A)| = 256.